S Germanis

P Atkinson

R Hostein

S Suffit

F Margaillan

V Voliotis

B Eble

Electrical control of optically pumped electron spin in a single GaAs/AlAs quantum dot fabricated by nanohole infilling

We demonstrate here electrical control of the sign of the circularly polarized emission from the negatively charged trion, going from co-to contrapolarized with respect to the circular polarization of the laser, using a GaAs/AlAs quantum dot (QD) embedded in a field effect structure. The voltage range over which the trion is negatively (contra) circularly polarized is shown to be dependent on the laser excitation energy within the P-shell resonance. The negative polarization never exceeds ∼ -15%, in stark contrast to measurements on InAs/GaAs QDs reported by M. E. Ware et al. [Phys. Rev. Lett. 95, 177403 (2005).] in which a negative polarization reaching -95% was observed. This result is shown to be a consequence of the low-symmetry confinement potential of these GaAs/AlAs QD, which are fabricated by partial infilling of asymmetric droplet-etched nanoholes. This low QD symmetry also leads to optical activity of the dark spin configuration of the triplet state, which we measure experimentally by photoluminescence excitation spectroscopy. A simple, semiquantitative model explaining both the optical activity of the dark spin configuration and the maximum degree of negative polarization is presented.

I. INTRODUCTION

An electron spin trapped in a semiconductor quantum dot (QD) is a robust qubit whose quantum state can be fully manipulated by laser sequences tuned to the lowest excited state, the negatively charged trion. In this case, an applied external magnetic field is imperative to lift the spin degeneracy of the trion state to create a system and initialize the spin by optical pumping techniques [1]. In contrast, the trion's excited state (hot trion), which is formed with one electron on the S orbital and one electron on the P orbital, gives rise to triplet states whose degeneracy is lifted by the electron-hole exchange interaction. As the total electronic spin within the negatively charged trion X -is zero, the circular polarization of the luminescence maps the spin of the hole which recombines. When the circular polarization is negative, i.e., contrapolarized with respect to the laser excitation, an efficient flip-flop process occurs between one of the electrons and the photocreated hole spin. The spin flip flop of the electron and hole, arising before the hot trion can decay to the trion ground state, leads to initialization of the spin of the electron remaining in the QD [2]. This negative polarization effect has been used in measurements of an ensemble of n-doped QDs to probe the dynamics of the coupled electron-nucleus spin system in the low-magnetic-field limit [3][4][5].

The degree of circular polarization, defined as P c = (I σ + σ + -I σ + σ -)/(I σ + σ + + I σ + σ -), has been shown to reach a value close to -100% for the negatively charged trion of a * germanis@insp.jussieu.fr † benoit.eble@insp.jussieu.fr single InAs/GaAs QD [6,7]. Here, we study the dynamics of the negative circular polarization from a strain-free, highly asymmetric single GaAs/AlAs QD grown by partial infilling of an in situ droplet-etched nanohole [8]. These QDs were embedded in a n-i-Schottky diode structure, designed to permit discrete charging of QDs with a wide voltage range where the X -state was dominant.

We investigate here the combined effect of applied bias and excitation energy on the degree of negative circular polarization of the trion emission. While optically addressing the hot trion through P-shell excitation of the QD, we observe that the applied bias can be used to control the degree of negative circular polarization up to its complete disappearance. Similarly, tuning the excitation energy within the broad Pshell resonance can lead to suppression of either the negative or positive circular polarization. In addition, we carry out resonant excitation of the hot trion by exciting the forbidden S-P transition, involving the photocreation of an electron in the conduction band P orbital and a hole in the lowest-energy valence band S-orbital. In this configuration we observe three lines in the photoluminescence excitation (PLE) spectra of the hot trion, corresponding to the two bright triplet states and the dark spin configuration. This is in contrast to measurements of InAs/GaAs QDs, where only selective absorption of the two bright triplet states has been observed [6]. We explain the optical activity of the dark spin configuration of the hot trion by considering the anisotropic morphology of the nanohole which determines the shape of the GaAs/AlAs QDs studied in this work. This anisotropy lowers the group symmetry of the confinement potential down to C s , which strongly modifies the optical activity of dark spin configuration excitons [9]. We develop a simple model where the electron-hole exchange interaction is expressed in the reduced-symmetry framework and show that it agrees well with the experimental results.

II. EXPERIMENTS

The sample used consisted of GaAs/AlAs QDs fabricated by nanohole infilling inside a n-i-Schottky structure with a 168-nm-wide intrinsic region. These nanoholes, created by Al droplet deposition followed by annealing under As 4 , are typically 8-10 nm deep and ∼60 nm (∼90 nm) wide along [-110] ([110]). Notably, the deepest point of the hole is generally offset from the hole center due to droplet motion during etching, leading to asymmetry of the hole profile along the [-110] direction [10]. The nanoholes are overgrown by a 1-nm-thick GaAs layer, but net migration towards the bottom of the hole during a short 15-s interrupt leads to the dots studied here being 2-4 nm high. Nanohole etching occurs after an 18-nm Al 0.45 Ga 0.55 As intrinsic spacer has been grown over the n-doped contact, leading to dots being located nominally around 10 nm from the n contact. However, variation in both nanohole size and infilling amount across the 2 inch wafer leads to a range in dot size and separation from the n contact. In this sample, the QD density is very low (one or two QDs per square micrometer), allowing individual QDs to be easily isolated for optical spectroscopy experiments. The QD charge is controlled by applying a bias between the semitransparent Schottky gate evaporated on the top surface of the sample and the diffusive Ohmic contact. The different charged states for a typical QD emitting at 1.74 eV are revealed in Fig. 1(a), which shows the evolution of the photoluminescence signal when sweeping the bias. Both the positive (X + ) and negative (X -) trions are clearly identified and are well separated from the neutral exciton (X 0 ) voltage domain.

Figure 1(b) shows the photoluminescence excitation (PLE) spectra for the three excitonic charge states of the same QD. The P-shell transition and the D shell are located ∼26 and ∼34 meV above the lowest-energy transitions. These values agree well with the performed eight-band k • p calculations of a QD with a 50-nm radius and a 4-nm height. The electronic excitation resonances are several meV wide, although their FIG. 2. Fine structure of the hot trion. The lifted degeneracy of the triplet states is fully shown when the hole occupies the S-orbital level, whereas only T 0 and T -1 are represented when the hole is on the P-like orbital. All the states of the hot trion are twofold degenerate depending on the hole spin projection. As the QD is excited by a σ + polarized laser, only the states with hole spin up are depicted. ˜ Se-Pe and ˜ S h -Pe are the energy differences between the orbital levels renormalized by the Coulomb interaction. The blue path gives rise to contrapolarized luminescence of the trion, whereas the red path leads to copolarized trion emission.

width decreases going from X + to X 0 to X -, i.e., as the bias approaches the flat-band conditions, here close to 1 Vt. This indicates that carrier tunneling contributes significantly to the broadening. However, even under strong forward bias, when the band structure is close to the flat-band condition, the resonances associated with the negative trion X -remain broad and larger than ∼3 meV. We interpret this as a consequence of the very fast hole relaxation time (τ P h →S h ∼ 1 ps) from the P to the S orbital.

To optically address the hot trion, we first choose to excite the P-shell transition of the QD while polarizing the diode in the voltage range where the X -is stable. The different configurations of charge carriers in the dot following this excitation and subsequent carrier relaxation and recombination are shown in Fig. 2 [11]. For simplicity, we consider only the lowest-energy P orbital (let's say P x ) for the valence and the conduction band, and we discard the influence of the splitting between P x and P y envelope wave functions that comes from Contrary to the relaxed trion where all the carriers are in the S shell and the electron-hole exchange interaction plays no role, for the hot trion the isotropic component of the electronhole exchange interaction lifts the degeneracy of the triplet by an amount ˜ 0 = 1 2 ( S e -S h 0

+ P e -S h 0 ) [ ˜ 0 = 1 2 ( S e -P h 0 + P e -P h 0
)] if the hole occupies the S-like orbital (P-like). The quantity ˜ 0 is expressed using the effective electron-hole exchange interaction Hamiltonian for C 2v symmetry [START_REF] Ivchenko | Superlattices and Other Heterostructures: Symmetry and Optical Phenomena[END_REF]14]:

Ĥeeh = i=(S e ,P e ) -2 i-S h 0 Ŝe i z Ĵh z + δ i-S h 1 2 ( Ŝe i + Ĵh -+ Ŝe i - Ĵh + ), ( 1
)
where Ŝe i is the electron spin operator occupying the i orbital and Ĵh is the pseudo-1/2 spin operator for the heavy hole with the convention Ĵh

z | ⇑ = 1 2 | ⇑ and Ĵh z | ⇓ = -1 2 | ⇓ .
We neglect in Eq. ( 1) the anisotropic exchange splitting of the dark exciton δ 2 as it is more than one order of magnitude weaker than the exchange splitting of the bright exciton δ 1 .

When the laser is σ + polarized and tuned to the P h -P e transition energy, i.e., 27 meV above the trion emission energy, the bright triplet states |T -1 , ⇑ P h and |T 0 , ⇑ P h are populated. They decay rapidly in a typical hole relaxation time τ P h →S h to |T -1 , ⇑ S h and |T 0 , ⇑ S h . A Pauli spin blockade prevents the P electron of the |T -1 , ⇑ S h state from relaxing to the S shell without an electron-hole spin flip flop. This electron-hole spin flip flop is due to the anisotropic part of the electron-hole exchange interaction [see Eq. ( 1)]. As a consequence, |T -1 , ⇑ P h generated by σ + polarized excitation relaxes via the intermediate |T -1 , ⇑ S h state to the |S 0 , ⇓ S h trion state. As shown by the blue path drawn in Fig. 2, this relaxation pathway ultimately leads to emission from the negative trion X -which is contrapolarized with respect to the laser excitation.

The efficiency of the electron-hole spin flip-flop mechanism which leads to contrapolarized emission can be measured by time-resolved photoluminescence experiments [15]. Figure 3(a) shows the dynamics of the co-and contrapolarized components of the X -photoluminescence following a 2-ps circularly polarized laser pulse. The laser energy is 1.761 eV, i.e., within the P h -P e transition, and the applied bias is 0.78 V. The PL signal is recorded by a Si avalanche photodiode (APD) with a Gaussian time response at FWHM of 240 ps (see Appendix A). After 200 ps, the contrapolarized emission intensity is greater than the copolarized one, indicating that the degree of circular polarization P c becomes negative. The inset in Fig. 3(a) shows a zoom for a time window centered at 200 ps. We observe that the maximum intensity of the copolarized intensity is reached at ∼160 ps, whereas for contrapolarized intensity, the maximum is located at ∼220 ps. The contrapolarized intensity delay time is clearly associated with the flip-flop process effective time τ f f (see Fig. 2). Using a simple rate equation model and taking into account the APD time response, we estimate that τ f f = 70 ps (see Appendix B). This value is about four times faster than the contrapolarized PL component delay time observed in InAs [2] and InP QDs ensembles [15].

In more detail, three mechanisms, reliant on the electronhole anisotropic exchange interaction, have been invoked to explain the relaxation of the Pauli-blocked |T -1 , ⇑ S h state to the |S 0 , ⇓ S h trion state [6]. The first mechanism relies on a combination of electron-hole and electron-phonon interactions to couple |T -1 , ⇑ S h to the virtual state |S 0 , ⇓ S h +1LO . By treating this latter interaction in the framework of the Fröhlich model, it was shown in Ref. [7] that this mechanism, which successfully described the negative circular polarization dynamics of the trion of a single InAs QD, becomes ineffective for a large mismatch between the S-P splitting energy and the LO phonon energy [16]. This is also the case for the GaAs/AlAs dots studied here, where the S-P splitting energy is ∼16 meV (see Fig. 2), compared to the 36-meV LO phonon energy in GaAs. The second mechanism is based on coupling between the |T -1 , ⇑ S h and |S 0 , ⇓ S h states, followed by thermalization to the X -(|S 0 , ⇓ S h ) state [2]. In this case, the spin flip-flop rate τ -1 f f is proportional to

(δ S h -Se 1 -δ S h -Pe 1 ) 2 ( ee -˜ 0 ) 2 .
Since the electron exchange interaction energy ee is in the meV range, τ f f has been estimated to be of the order of nanoseconds [7,17]. This is much longer than the value of τ f f experimentally observed here. In contrast, the third mechanism is based on coupling between the |T -1 , ⇑ S h and |T 0 , ⇓ S h states followed by thermalization to the |S 0 , ⇓ S h state [6]. In this case, τ -1 f f is proportional to

(δ S h -Se 1 +δ S h -Pe 1 ) 2 ˜ 2 0 .
Since ˜ 0 is of the order of 0.1 meV, this will give a much shorter flip-flop lifetime than the previous mechanism. The subsequent thermalization of |T 0 , ⇓ S h to the X -state was found to be ∼10 ps [6], only one order of magnitude greater than for |S 0 , ⇓ S h , due to a spin-orbit-induced asymmetric electron-electron interaction between |T 0 , ⇓ S h and |S 0 , ⇓ S h [6,18,19]. This relaxation mechanism is therefore compatible with the experimental determination of τ f f of 70 ps here.

Figure 3(b) shows the dynamics of the degree of circular polarization of X -emission. The experimental points are plotted using the data shown in Fig. 3(a), i.e., without any deconvolution procedure. We observe a long decay time of the negative circular polarization. From this decay time we evaluate the effective hole spin lifetime τ h s to be 2.7 ns (see Appendix B). This is the first time that the hole spin lifetime has been measured in such GaAs/AlAs QDs and indicates that it is relatively short. For comparison, similar measurements of the InAs/GaAs QD ensemble gave τ h s 20 ns [20]. Figures 3(c) and 3(d) show the effect of a small change in applied bias on the X -polarization while keeping the same laser excitation energy. With an applied bias of 0.87 V, there is no delay between the contra-and copolarized components, with both having their maximum intensity at ∼180 ps, as shown by the vertical dashed line in Fig. 3(c). Figure 3(d) shows that this lack of delay means that there is no negative degree of circular polarization at this bias voltage.

At this stage, two conclusions arise from these observations. First, since increasing (decreasing) the bias is able to decrease (increase) the relative fraction of contrapolarized emission, it must be inducing a weakening (enhancing) of the relative absorption of the |T -1 , ⇑ P h state compared to the |T 0 , ⇑ P h state. Second, there is no measurable delay between the contrapolarized and copolarized emissions for a bias voltage of 0.87 V (compared to the 70-ps delay of the contrapolarized emission at 0.78 V). The fraction of emission that is contrapolarized, however, remains significant [Figs. 3(c) and 3(d)], with an ∼ + 20% positive degree of circular polarization. The lack of delay time indicates that this contrapolarized emission is not due to the spin flip-flop relaxation of the |T -1 , ⇑ S h state to the |S 0 , ⇓ S h state. A fast hole spin relaxation time in the P orbital would, however, lead to rapid filling of both the |T 0 , ⇑ P h and |T 0 , ⇓ P h states after excitation with positive circularly polarized light [21], providing a possible route to lower the degree of circular polarization. This hole spin relaxation in the P shell prior to thermal relaxation to the S shell would also contribute to decreasing the amplitude of the negative degree of circular polarization at 0.78 V [see Fig. 3(a)]. Under this voltage condition, where the |T -1 , ⇑ P h state is being filled preferentially to the |T 0 , ⇑ P h state, hole spin relaxation in the P orbital would reduce the fraction of the |T -1 , ⇑ P h state due to filling the |T -1 , ⇓ P h state. This would subsequently reduce the contrapolarized emission that is due to the spin flip flop of the |T -1 , ⇑ S h state followed by thermal relaxation of the |T 0 , ⇓ S h state.

This assumption of fast hole spin relaxation in the P orbital is used in the calculations presented in Appendix B, where τ f f is quantified as 70 ps, taking into account the APD temporal response.

To study further the effect of applied bias on the relative absorption of the different triplet states, Fig. 4(a) shows the X -degree of circular polarization using a cw spectrally narrow laser tuned to slightly different energies within the P-shell electronic resonance. For a fixed excitation energy, the degree of circular polarization P c changes from negative to positive (contrapolarized to copolarized) with increasing bias voltage. However, with increasing laser energy, the voltage range for which P c is negative increases, while both the voltage range and the magnitude of positive P c decreases. The critical excitation energies for which P c will remain positive (negative) over the entire voltage range of the X -emission are found experimentally to be E Laser -E X -22 meV ( 30 meV).

The combined effect of excitation energy and applied bias on the presence or absence of a negative degree of circular polarization can be explained by considering the Stark shift of the QD energy levels with applied bias and the induced effect on the probability of absorption of different triplet states. This is shown schematically in Fig. 4(b). When the σ + polarized laser energy is high enough, the absorption on the higherenergy |T -1 , ⇑ P h state will be greater than that of the |T 0 , ⇑ P h state (see Fig. 2). Fast relaxation from the |T -1 , ⇑ P h state to the |T -1 , ⇑ S h state leads to contrapolarized X -emission, as displayed by the blue arrows in Fig. 4(b). Increasing the bias leads to a decrease of the electric field across the dots and a blue Stark shift of the optical transitions [22]. When the Stark shift is sufficiently large to bring the |T 0 , ⇑ P h state closer into resonance with the excitation laser energy than the |T -1 , ⇑ P h state, absorption of the |T 0 , ⇑ P h state will dominate. Relaxation from this state leads to copolarized emission of X -, as shown by the red lines in Fig. 4(b).

Because of the large broadening of the P-shell resonance [Fig. 1 (b)], the transition from absorption in the |T -1 , ⇑ P h state to the |T 0 , ⇑ S h state when changing the laser energy is not abrupt.

To illustrate this, Fig. 4(c) shows the circular-polarizationresolved PLE over a small spectral window around the transition from positive to negative P c at fixed bias (0.825 V). When the laser is tuned to the lower-energy side, P c is positive, whereas on the higher-energy side, P c is negative, which is consistent with the higher energy of T -1 compared to T 0 . The sign of P c crosses from positive to negative at E Laser -E X -∼ 26 meV. This corresponds to the point where absorption and recombination of the T -1 and T 0 states are balanced, giving similar co-and contratrion intensities. If the laser energy is set too far above or below this position, the amplitude of the Stark shift over the voltage range of the stability of X - is not enough to change the absorption from T -1 to T 0 (or vice versa), and there will subsequently be no change in sign of P c on changing the bias. This situation is illustrated in Fig. 4(a) for E Laser -E X -= 22 meV and, to some extent, for E Laser -E X -= 30 meV.

When looking carefully at Fig. 4(a), we observe that the maximum positive P c reaches a value greater than +25%, whereas the maximum negative P c seems to be pinned at ∼ -15%. Under excitation and bias conditions where absorption can be assumed to be predominantly either in the |T -1 , ⇑ P h or |T 0 , ⇑ P h state, P c of X -will reflect spin relaxation occurring as the hot trion states relax down to either the |S 0 , ⇓ S h or |S 0 , ⇑ S h state (Fig. 2). If P c were being limited by spin relaxation of the hole in the P orbital prior to its relaxation to the S shell, then this would lead to a reduction of the maximum P c for both co-and contrapolarized components of the PL in a similar way.

To understand the lower amplitude for the negative P c , we set the laser energy to the position of the S h -P e transition while recording the PL signal from the X -line. In principle, the S h -P e optical transition is not allowed due to the parity of the envelope wave functions; however, the low symmetry of these QDs fabricated by nanohole infilling efficiently relaxes the optical selection rules [START_REF] Ediger | Physics of Semiconductors: 27th International Conference on the Physics of Semiconductors -ICPS-27[END_REF].

Figure 5(a) shows the PLE for X -while the linearly polarized laser scans the S h -P e transition ∼16 meV above the trion energy. The laser intensity was reduced as much as possible while still maintaining an acceptable signal-to-noise ratio. The amplitude of the PLE signal is seven times weaker than the signal obtained when exciting the P-shell resonance [Fig. 1(b)]. We observe a complex multipeak structure appearing for E Laser -E X -∼ 16.5 meV. This can be fitted satisfactorily by three equidistant Lorentzian lines which correspond to the absorptions of |T -1 , ⇑ S h and |T +1 , ⇓ S h at high energy (in blue in Fig. 5), |T 0 , ⇑ S h and |T 0 , ⇓ S h (in red), and the "dark" hot trion configurations |T +1 , ⇑ S h and |T -1 , ⇓ S h (in black) at low energy. The additional, well-separated, weak electronic resonance observable at 17.5 meV is assigned to absorption of the singlet states |S 0 , ⇑ S h and |S 0 , ⇓ S h . This result differs from similar measurements performed on InAs/GaAs QDs where only the bright triplet states were observed [6]. Before discussing in detail the reason for the appearance of the hot trion dark state configuration, we first here confirm the consistency of our interpretation.

(i) From the assignment of the peaks in Fig. 5 we are able to quantify ˜ 0 = 135 μeV [START_REF]The value of ˜ 0 is consistent with the dark-bright exciton splitting measured by magnetospectroscopy carried out in the Voigt configuration on similar undoped samples[END_REF] and ee = 1.1 meV. The magnitude of the splitting ee between T 0 and S 0 for this GaAs/AlAs QD is confirmed by PL spectroscopy measurements on the same dot where ee ∼ 1 meV is evaluated from the separation of the singlet-triplet components of the X 2- state (see Appendix A). It should be noted that this value is much weaker than in InAs/GaAs QDs, where it lies in the range of ∼5meV [6,[START_REF] Urbaszek | [END_REF].

(ii) The ratio of the oscillator strength between the two bright triplet states r =

f |T -1 ,⇑ S h f |T 0 ,⇑ S h
is expected to be equal to 2 [6,16,26,27]. We find that the ratio of the integrated area of blue and red Lorentzian curves is 2.2, in good agreement with the expected value. As pointed out in Ref. [6], these absorption peaks are broadened when the laser power is enough to saturate the transitions. By repeating the same experiments shown in Fig. 5(a) using an excitation power two times more intense (not shown), we measure the same width of the PLE lines, indicating that they are already close to the saturation regime in Fig. 5(a). As a consequence, the linewidths of the Lorentzian fits are much greater than the natural linewidth due to laser-power-induced broadening, which explains the overlap of the different lines.

(iii) Circular-polarization-resolved PLE obtained when the laser is σ + polarized [Figs. 5(a) and 5(b)] confirmed that only the PLE line associated with the |T -1 , ⇑ S h state is contrapolarized with P c = -10%. To calculate the degree of circular polarization we use here the integrated areas from the σ + σ + and σ + σ -PLE spectra, i.e., I σ + σ + and I σ + σ -. The same analysis gives a circular polarization of +30% for both T 0 and the dark PLE lines and a value of +10% for S 0 . The circular polarization of S 0 is surprisingly weak given the long hole spin relaxation time (2.7 ns), although the positive sign is the expected one. It is possible that the large background in the PLE signal due to the high power needed for the measurement is leading to an underestimation of this value. Note that similar experiments performed on InAs QDs have shown that the degree of circular polarization obtained for the |T -1 , ⇑ S h PLE line exceeds -95%, in stark contrast to our result. We show next that the modest degree of negative circular polarization and the presence of the dark hot trion PLE line are strongly related.

The optical activity of the dark spin configuration exciton results from the low C s symmetry of the QD confinement potential [9,28,29]. This symmetry breaking is taken into account in the effective Hamiltonian by adding new contributions to Ĥeeh in Eq. (1) [9,28] as

δ ĤC s eeh = i=(S e ,P e ) δ i-S h 11 Ĵh z ( Ŝe i + + Ŝe i -) + δ i-S h 12 Ŝe i z ( Ĵh + + Ĵh -), (2)
where δ i-S h 11 and δ i-S h 12 are the energies related to the noncollinear terms coupling bright-to-dark excitonic configurations by flipping solely the spin of the electron occupying the orbital i or hole on the S h orbital. Treating similarly δ ĤC s eeh and the anisotropic part of Ĥeeh , perturbation theory gives, to first order, the dark hot trion state as

| T +1 , ⇑ S h = |T +1 , ⇑ S h -α|T 0 , ⇑ S h , (3)
and for the higher-energy triplet state it gives ]. In Eqs. ( 3) and ( 4), we have kept the dominant terms in ˜ -1 0 and have neglected terms with energy denominators 2 ˜ 0 [START_REF]Se-S h 12 + δ Pe-S h 12[END_REF] and ee from the perturbative expansion. Equation (3) shows that the dark triplet state is mixed with the bright |T 0 , ⇑ S h , with a coupling α transferring oscillator strength to the dark configuration. Equation (3) also predicts that the dark state should have the same circular polarization sign as the |T 0 , ⇑ S h state, in agreement with the experimental observations from the PLE measurements. As a consequence, α 2 quantifies the ratio of the oscillator strengths of the bright and dark states, which can be evaluated to first order as the ratio of the integrated areas of the PLE lines in Fig. 5(a Equation ( 4) describes the effect of the QD asymmetry on the |T -1 , ⇑ S h state. The first term gives the coupling between |T -1 , ⇑ S h and |T 0 , ⇓ S h . This coupling is governed by the anisotropic exchange splittings present in β and is the origin of the negative circular polarization. The second term describes an additional coupling to |T 0 , ⇑ S h , weighted by α. This term comes from including δ ĤC s eeh , and it acts to offset the coupling to the |T 0 , ⇓ S h state, thereby reducing the value of the negative P c . In other words, when the condition β 2 α 2 is fulfilled, the negative circular polarization could be close to -100%, as already reported for InAs/GaAs QDs [6], but as α becomes more significant, the negative circular polarization is reduced.

| T -1 , ⇑ S h = |T -1 , ⇑ S h + β|T 0 , ⇓ S h + α|T 0 , ⇑ S h , (4)
The maximum achievable negative polarization measured in cw experiments can be expressed by

P lim c = α 2 -β 2 α 2 +β 2 .
In order to evaluate P lim c , α can be experimentally determined from the relative PLE intensity of the dark and bright trion states, as previously discussed. Estimation of β 2 can be carried out by using the simplification δ P e -S h 1 ∼ δ S e -S h 1 , with δ S e -S h 1 being the fine-structure splitting of the neutral exciton measured to be 95 μeV (see Appendix A). We find then β 2 = 0.24 and thus P lim c = -30%, which is confirmed experimentally. The validity of the expression defining P lim c relies on the fact that the splittings between T -1 and T 0 with hole spin both up and down are the same, and the thermalizations of both T 0 towards the ground trion states are identical. Taking into account all the perturbative terms in Eq. ( 4) would lead to nontrivial expression of P lim c and may explain the difference between the theoretical value of P lim c and the maximum experimental negative degree of polarization of ∼ -15%. It should be noted that the limitation of the degree of negative circular polarization is linked to the nanohole etching conditions used, which yielded relatively deep and wide nanoholes that had an asymmetric profile along [-110] and that were only partially infilled. Other work has shown that it is possible to have high-symmetry GaAs/AlAs QDs in nanoholes with the use of smaller initial droplets, leading to less deep nanoholes, annealing under an As 2 overpressure, and complete hole infilling with GaAs [START_REF] Huo | [END_REF]. This means that further optimization of the diode structure and dot growth conditions should improve the degree of negative circular polarization observed here.

This simple model, based on the effect of symmetry breaking of the QD confinement potential, explains simultaneously the appearance of absorption of the dark spin configuration hot trion and the reduction of negative circular polarization of X -.

III. CONCLUSION

In this paper, we demonstrated electrical control of the exciton charge state of a GaAs/AlAs quantum dot fabricated by in situ droplet etching of nanoholes and partial infilling. Using P-shell excitation, electrical control of the degree of circular polarization of the X -state, going continuously from positive (copolarized) to negative (contrapolarized), was demonstrated. This control over the sign of the degree of circular polarization is due to a modification of the relative occupation of the hot triplet states either by the Stark shift or by the laser excitation energy within the P-shell resonance. This voltage dependence means that modulation of the bias component, while remaining within the stability region of the negative trion, could be used to provide on-demand control of the sign of the circular polarization from a single quantum emitter. This has potential application in quantum information protocols.

A higher degree of negative polarization can be achieved by optimization of the diode structure to increase the stability region of the negative trion together with improvement in the growth conditions during nanohole formation and infilling to give QDs with higher symmetry.

APD temporal response

The APD response is shown in Fig. 8(a) when a 2-ps laser pulse is sent to the detector. We extract then a Gaussian response G(t ) with a FWHM of 240 ps.

The intensity signal C(t ) measured by the detector is then equal to the convolution G(t ) [H (t )S(t )], where H (t ) is the Heaviside distribution and S(t ) is the deconvoluted PL signal.

Rate equation model for the PL dynamics

We present here the rate equation model used to fit the PL time dynamics in Fig. 2 when the laser is tuned to the P-shell resonance, considering the following reasonable assumptions: (i) the hole relaxation time from the P to S orbital is extremely fast, τ P h →S h ∼ 1 ps, (ii) the relaxation of the triplet T 0 towards the trion is much faster than τ f f for the triplet T -1 , (iii) all the bright triplet states with hole spin up or down can be populated after the hole thermalization due to the possible fast hole spin relaxation time when the hole occupies the P level [33] or due to the fact that the broad picosecond-pulse σ + laser can excite both |T -1 , ⇑ P h and |T 0 , ⇑ P h , and (iv) we neglect the role of the dark spin hot trion configuration.

This leads to The fitting parameters are τ f f , τ rad , and τ h s as well as the initial conditions which are susceptible to variations according to different applied biases used in Fig. 2. Finally, the PL dynamics detected by the APD is given by I σ + σ ± (t ) = 1 τ rad G(t ) [H (t )n |S 0 ,⇑ S h (⇓ S h ) (t ) ]. When fitting simultaneously the co-and contrapolarized intensities for biases of 0.78 and 0.87 V, we find then τ f f = 70ps, τ rad = 380ps and τ h s = 2.7ns. We present the deconvolution of the dynamics in Figs. 8(b) and 8(c).

FIG. 1 .

 1 FIG. 1. (a) Map of the photoluminescence of a single QD as a function of the applied bias. The laser energy is set at E Laser = 1.771 eV. (b)Fast-scan PLE for the three exciton states; -0.5, 0, and 0.7 V were applied to the Schottky diode to record the PLE spectra of X + , X 0 , and X -, respectively.

FIG. 3 .

 3 FIG. 3. (a) Time dynamics of the co-and contrapolarized luminescence of X -when the applied bias is 0.78 V. The laser energy is set 26 meV above the X -emission energy, corresponding to excitation of the P-shell resonance. The response of the APD to a 2-ps pulse is added in black. The red and blue lines are fits taking into account the detector convolution. (b) Dynamics of the circular polarization P c of the X -emission without deconvolution. (c) and (d) As in (a) and (b) but with applied bias of 0.87 V. the QD elongation perpendicular to the growth axis [12]. The fine structure of the hot trion, where one electron occupies an S orbital and a second electron occupies the P orbital, consists of the triplet states T 0 (↑ s ↓ p + ↓ s ↑ p )/ √ 2, T +1 (↑ s ↑ p ), and T -1 (↓ s ↓ p ) and a singlet S * 0 state (↑ s ↓ p -↓ s ↑ p )/ √ 2. The singlet state is split from the triplet states by the electron-electron exchange interaction ee .Contrary to the relaxed trion where all the carriers are in the S shell and the electron-hole exchange interaction plays no role, for the hot trion the isotropic component of the electronhole exchange interaction lifts the degeneracy of the triplet by an amount ˜ 0 = 1 2 ( S e -S h

FIG. 4 .

 4 FIG. 4. (a) Voltage scans of the X -emission circular polarization for different excitation energies within the P-shell resonance. The black dashed lines are guides to the eye. (b) Schematic showing the change in polarized emission from contrapolarized (blue) to copolarized (red) on changing the electric field across the QD by F (change in applied bias V > 0). The electric-field-induced energy shift of the QD states relative to the exciting laser energy leads to a decrease in absorption of the |T -1 , ⇑ P h state compared to |T 0 , ⇑ P h (see the text for a detailed explanation). (c) Circular-polarizationresolved PLE of X -with a laser step of 11 μeV. The dashed red and blue lines are guides to the eye. The apparent noise in P c comes from fluctuations in the electric field at this applied voltage, observable as fluctuations in the photocurrent during the PLE measurement.

FIG. 5 .

 5 FIG. 5. (a) X -PLE spectra when the laser energy is scanning the S h -P e transition. The laser is linearly polarized and no-polarization selective detection is employed. The PLE lines are fitted by including a local quadratic background shown by the purple dashed curve. (b) and (c) The same experiment setting the laser polarization σ + and analyzing (b) σ + and (c) σ -. The laser intensity in (b) and (c) is twice that used in (a) to enhance the signal-to-noise ratio when measuring the circular polarization of the hot trion triplet components.

with α = 1

 1

11 ∼

 11 ) and is equal to 0.13. Assuming δ Se-Sh 11 ∼ δ Pe-Sh 11 , we obtained δ Se-Sh 70 μeV, in very good agreement with magnetospectroscopy experiments performed on similar QDs [31].

FIG. 8 .

 8 FIG. 8. (a) Gaussian response of the APD to a 2-ps laser pulse excitation. The red curve is a fit. We discard in the calculations the weak discrepancy at long-delay positive time where the intensity is very weak. (b) and (c) Deconvoluted signals corresponding to the experimental data presented in Fig. 3.

  ṅ|S 0 ,⇑ S h = -n |S 0 ,⇑ S h τ rad -n |S 0 ,⇓ S hn |S 0 ,⇑ S h 2τ s h + n |T +1 ,⇓ S h τ f f , ṅ|S 0 ,⇓ S h = -n |S 0 ,⇓ S h τ rad -n |S 0 ,⇑ S hn |S 0 ,⇓ S h 2τ s h + n |T -1 ,⇑ S h τ f f , ṅ|T +1 ,⇓ S h = -n |T +1 ,⇓ S h τ f f , ṅ|T -1 ,⇑ S h = -n |T -1 ,⇑ S h τ f f ,where τ rad is the radiative lifetime of the trion.The initial conditions according to assumptions (i) and (ii) are defined asn |S 0 ,⇑ S h (t = 0) = n |T 0 ,⇑ P h (t = 0), (B1)n |S 0 ,⇓ S h (t = 0) = n |T 0 ,⇓ P h (t = 0), (B2) n |T -1 ,⇑ S h (t = 0) = n |T -1 ,⇑ P h (t = 0), (B3) n |T +1 ,⇓ S h (t = 0) = n |T +1 ,⇓ P h (t = 0),(B4)andn |T +1 ,⇓ P h (t = 0) + n |T -1 ,⇑ P h (t = 0)+ n |T 0 ,⇓ P h (t = 0) + n |T 0 ,⇑ P h (t = 0) = 1. (B5)

ACKNOWLEDGMENTS

This work was financially supported by the French National Research Agency (ANR "SPINEX" ANR-17-CE30-0022-01 and ANR "ISQUAD" ANR-18-CE47-0006-01), by the Paris Île-de-France Région in the framework of DIM SIRTEQ, and by the Cluster of Excellence MATISSE led by Sorbonne Universités. We thank C. Manquest for technical support.

APPENDIX A: EXPERIMENTAL ESTIMATION OF ee FOR A SINGLE GaAs/AlAs QD FABRICATED BY NANOHOLE INFILLING

We present here the experiments which provide quantitative measurements of the energy splittings used in the main text. All the experiments were performed on the same QD used previously with green laser excitation at a temperature of 4 K.

When the green laser excites the QD, new lines associated with a highly charged exciton appear when the applied bias is inside the domain of existence of X -(see Fig. 1). In order to confirm the estimation of ee done using PLE experiment via the S h -P e transition, the optical spectroscopy of the doubly charged exciton stands as a good tool for estimating ee , FIG. 6. Photoluminescence spectrum from the studied QD showing the fine structure of X 2-. provided both triplet and singlet components can be detected [START_REF] Urbaszek | [END_REF]. In Fig. 6, we distinguish the weak PL associated with the singlet appearing on the low-energy side of the charged biexciton 2X -, as well as the two triplet components. This provides an estimation of ee of 1 meV, in good agreement with the experimental determination given in Fig. 5(a).

Also, in Fig. 7 we show the experimental result leading to the estimation of δ S e -S h 1 . We analyze the neutral exciton X 0 emission energy position as a function the linear polarization by rotating a λ/2-wave plate placed before the linear polarizer and from the amplitude of the sinusoidal fit we extract δ S e -S h 1 of ∼95 μeV.

APPENDIX B: ANALYSIS OF THE CIRCULAR POLARIZATION DYNAMICS

The objective of this Appendix is to present a consistent determination of τ f f giving the experimental result in Fig. 3. This requires, first, calculating the PL dynamics using a simple rate equation model [15] and, second, taking into account the time response of the APD.


Alice Vilotte

Hugues Bodiguel

Komla Ako

Deniz Z Gunes

Christophe Schmitt

Clément De Loubens

Kinetic and structural characterization of whey protein aggregation in a millifluidic continuous process

Keywords: Heat-induced aggregation, Unfolding, Kinetics, Flow, SAXS

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

Introduction

Whey protein isolates (WPI) are of interest because of their nutritional and functional properties in food applications. Heatinduced aggregation coupled with the process conditions of whey proteins gives them new functional properties that can be used to impart the specific structural and physical properties of food products. Therefore, a thorough understanding and control of the aggregation process is required in order to design specific functional whey protein aggregates. The heat-induced aggregation process of whey protein (WP) is a complex multi-stage process but despite numerous studies, there are still many open questions that have not yet been resolved. [START_REF] De La Fuente | Recent advances in the characterisation of heat-induced aggregates and intermediates of whey proteins[END_REF][START_REF] Nicolai | β-lactoglobulin and wpi aggregates: Formation, structure and applications[END_REF][START_REF] Mezzenga | The self-assembly, aggregation and phase transitions of food protein systems in one, two and three dimensions[END_REF].

The rate of denaturation and aggregation of WPI depends on several factors such as pH, ionic strength, concentration and temperature. The measurement of apparent reaction rates of βlactoglobulin and α-lactalbumin denaturation (the major component of WPI) on a wide range of temperature (70 -150 • C) showed an abrupt change of the rate constant at 90 and 80 • C, respectively [START_REF] Dannenberg | Reaction kinetics of the denaturation of whey proteins in milk[END_REF]. Subsequent studies have found that the overall aggregation process was limited by unfolding at low temperature (< 85 • C) and aggregation reaction was the rate limiting step in the upper temperature range * hugues.bodiguel@univ-grenoble-alpes.fr (> 100 • C) [START_REF] Verheul | Kinetics of heatinduced aggregation of β-lactoglobulin[END_REF][START_REF] Spiegel | Whey protein aggregation under shear conditions-effects of lactose and heating temperature on aggregate size and structure[END_REF]. Beyond kinetics studies, conformational changes of WPI have also been investigated to explain the rates of denaturation and aggregation [START_REF] Relkin | Thermal unfolding of β-lactoglobulin, α-lactalbumin, and bovine serum albumin. a thermodynamic approach[END_REF][START_REF] De Wit | Thermal behaviour of bovine β-lactoglobulin at temperatures up to 150 • c. a review[END_REF]. While most of these studies did not make a distinction between unfolding limited from nucleation limited reaction, general mechanistic descriptions of heat-induced aggregation of WPI recognize an initiation, a propagation and a termination step [START_REF] Roefs | A model for the denaturation and aggregation of β-lactoglobulin[END_REF][START_REF] Tolkach | Reaction kinetic pathway of reversible and irreversible thermal denaturation of β-lactoglobulin[END_REF] by analogy with polymer radical chemistry [START_REF] Roberts | Non-native protein aggregation kinetics[END_REF][START_REF] Morris | Protein aggregation kinetics, mechanism, and curve-fitting: A review of the literature[END_REF]. More specifically, heat-induced aggregation of WPI involves the reversible self-association of two or more "reactive" monomers to form reversible non-native oligomers. The unfolding of these chains creates the smallest net irreversible aggregate, the nuclei, from which aggregates grow by covalent and/or noncovalent interactions and condensate. One main limitation of the kinetics studies cited above is that they based their analysis on the consumption of native proteins undergoing aggregation in order to measure the global kinetics constant, whereas it would also be valuable to measure the concentration of aggregated proteins to precisely determine the mechanism which limits the reaction. In this regard, X-ray scattering techniques can be used to simultaneously follow the concentration of native and aggregated proteins [START_REF] Pérez | Heatinduced unfolding of neocarzinostatin, a small all-β protein investigated by small-angle x-ray scattering[END_REF] and also to gain insight on the structural properties of the aggregates [START_REF] Moitzi | Structure of β-lactoglobulin microgels formed during heating as revealed by small-angle x-ray scattering and light scattering[END_REF].

The structure of aggregates can be modulated by varying the balance between repulsive / attractive forces [START_REF] Zuniga | Kinetics of formation and physicochemical characterization of thermally-induced βlactoglobulin aggregates[END_REF][START_REF] Nicolai | β-lactoglobulin and wpi aggregates: Formation, structure and applications[END_REF]. One striking feature is the transition from fibrillar aggregates to microgels and fractal aggregates by changing the pH from 2.0 to 7.0 [START_REF] Jung | Structure of heat-induced β-lactoglobulin aggregates and their complexes with sodium-dodecyl sulfate[END_REF]. The relationships of structural parameters of fractal aggregates with physico-chemical conditions and heat-treament have been well studied in regime of low temperature. Below 85 • C, aggregation is limited by unfolding and takes place over several hours [START_REF] Le Bon | Growth and structure of aggregates of heat-denatured β-lactoglobulin[END_REF][START_REF] Zuniga | Kinetics of formation and physicochemical characterization of thermally-induced βlactoglobulin aggregates[END_REF][START_REF] Nicolai | β-lactoglobulin and wpi aggregates: Formation, structure and applications[END_REF].

According to Le [START_REF] Le Bon | Growth and structure of aggregates of heat-denatured β-lactoglobulin[END_REF], the structure of fractal aggregates is independent of the concentration of β-lactoglobulin and the effect of the temperature is purely kinetic in the range 60 -87 • C. [START_REF] Zuniga | Kinetics of formation and physicochemical characterization of thermally-induced βlactoglobulin aggregates[END_REF] studied the effect of pH and holding time on the formation of thermally-induced β-lactoglobulin aggregates at 80 • C and showed that the size of the aggregates saturated for a holding time ranging from 50 to 200 s, depending on the pH (6.0 -6.8). However, these conditions are far from most industrial processes, in which a high temperatureshort time UHT treatment is applied, reaching close to 140 • C, and aggregation takes place over a few tens of seconds [START_REF] Dissanayake | Influence of heat and ph on structure and conformation of whey proteins[END_REF]. This rather fast aggregation is naturally combined with heat transfer in cases aggregation occurred in large-sized containers, such as batch reactors or test tubes. A simple way to increase the rate of heat transfer is to reduce the characteristic size of the sample down to the millimeter scale or below, as in a millifluidic continuous process [START_REF] Dannenberg | Reaction kinetics of the denaturation of whey proteins in milk[END_REF]. Coupling this kind of set-up with X-ray scattering techniques should also give us new insights into the kinetics of native and aggregated proteins, for example on the change of structural properties of the aggregates with the heating time. This kind of continuous process of agregation also offers an extra control parameter, which is the flow rate. It has been reported that the shear rate could affect the final aggregate size [START_REF] Spiegel | Whey protein aggregation under shear conditions-effects of lactose and heating temperature on aggregate size and structure[END_REF][START_REF] Simmons | The effect of temperature and shear rate upon the aggregation of whey protein and its implications for milk fouling[END_REF][START_REF] Wolz | Thermal aggregation of whey proteins under shear stress[END_REF][START_REF] De Guibert | Flow process and heating conditions modulate the characteristics of whey protein aggregates[END_REF], but this control parameter has not been widely studied and conclusions are conflicting.

Our objective was to study the kinetics of growth of WPI (4% w/w, 10 mM NaCl) at high temperature (92 • C) in a regime which should correspond to the transition between unfolding limited and aggregation-limited reactions [START_REF] Dannenberg | Reaction kinetics of the denaturation of whey proteins in milk[END_REF][START_REF] De La Fuente | Recent advances in the characterisation of heat-induced aggregates and intermediates of whey proteins[END_REF]. Structural characterizations were done by X-ray scattering and resonant mass measurement and also covered the length scales of both the aggregates and the proteins. Both methods gave us complementary insights about aggregation and unfolding mechanisms. In the same way as [START_REF] Dannenberg | Reaction kinetics of the denaturation of whey proteins in milk[END_REF] determined reaction rates of WPI at high temperature, we used a millifluidic continuous process to maximise the rates of heat transfer. This approach allowed us to accurately control the residence time, temperature history and shear rate.

Materials and methods

Materials and sample preparation

Whey protein isolate Prolacta 95 (batch 30916B) was purchased from Lactalis Ingredients (Bourgbarré, France). The composition of the WPI (Nx6.38) was (in g/100g powder wet basis): protein 88.5 (of which β-lacotglobulin 65%, α-lactalbumin 20%, soluble caseins 10% and almost 5 % of bovine serum albimin and immunoglobulin, ash 1.9, moisture 5.6, carbohydrates 0.14, fat 0.5). The mineral composition was (g/100g powder wet basis): Ca 0.35, Mg 0.06, Na 0.15, K 0.35, P 0.21, Cl 0.01.

WPI stock solution of 10% w/w was prepared by dispersing of the WPI powder in deionised water. Sodium Azide (0.02% w/w, Sigma-Aldrich CAS 26628-22-8) was added to the solution to prevent bacterial growth and gently stirred for 12 hours for complete dissolution of WPI powder. The stock solution was then dialyzed against salt-free deionised water containing 0.02 % w/w sodium azide. The concentration of the dialyzed solution of WPI was measured by UV absorption at 280 nm with an extinction coefficient of 1.186 L/g/cm. Finally, deionised water and sodium chloride was added to obtain a 4 % w/w WPI solution with an ionic strength of 10 mM, and pH was adjusted at 7.0 by the addition of sodium hydroxyde (1M).

Millifluidic continuous aggregation process

A continuous small scale process was designed to aggregate WPI solutions and decrease the characteristic time of heat transfer, so that aggregation occurred in isothermal conditions. The setup is presented in Fig. 1. WPI solution was injected at a constant flow rate Q with a syringe pump (low pressure neMESYS, Cetoni GmbH) at 22 • C through a PTFE tube of internal and external diameter d i = 0.81 mm (or d = 0.25 mm for the largest shear rates investigated, concerning the data displayed in Fig. 6) and d e = 1.59 mm, respectively. The tube was immersed in a stirred and heated water bath at temperature T ∞ and fixed at 92 • C for the entire study. The aggregates were collected for a few minutes in a tube immersed in ice and kept at 4 • C until analysis.

We studied the effects of the mean residence time t of WPI in the tube and the mean shear rate γ on the aggregation process. Both parameters were varied by changing the tube length L and the flow rate Q. They are defined by

t = πd 2 L 4Q (1) γ = 64 3π Q d 3 (2)
We investigated variations of t and γ from 20 s to 15 min and from 53 to 2666 s -1 . Consequently, Q and L ranged from 15 to 150 mL/h and 16.2 to 750 cm, respectively (d= 0.81 or 0.25 mm). The exact experimental conditions are given in the key of each figure. The flow was assumed to be laminar given that the range of Reynolds numbers, Re = du/ν was between 20.7 and 207, where ν = 3.2× 10 -7 m 2 /s the kinematic viscosity of water at 92 • C and u = 4Q/πd 2 the mean velocity in the tube.

The range of Peclet numbers investigated was always larger than 10 3 . Thus, the effective diffusion coefficient of proteins along the longitudinal axis of the tube was greater than the molecular coefficient because of the parabolic velocity profile

Figure 2: Reduced temperature (T -T ∞ ) / (T 0 -T ∞ )
as a function of the residence time t (Eq. 5) in a tube of 0.81 mm internal diameter for various Q (60 to 300 mL/h) and L (10 to 30 cm). The temperature of the water bath, T ∞ , was set at 80 • C (squares) and 90 • C (circles). The dashed line is the fit of Eq. 4 with the thermal characteristic time τ = 1.30 s (Eq. 5) that is independent of Q.

(i.e. Taylor-Aris dispersion, [START_REF] Taylor | Dispersion of soluble matter in solvent flowing slowly through a tube[END_REF]; [START_REF] Aris | On the dispersion of a solute in a fluid flowing through a tube[END_REF]; Cussler ( 2009)), and we expect a dispersion of the residence time of the proteins around the mean value t. Based on the scaling laws of [START_REF] Chatwin | The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe[END_REF][START_REF] Chatwin | The initial development of longitudinal dispersion in straight tubes[END_REF] and [START_REF] Latini | Transient anomalous diffusion in poiseuille flow[END_REF], we estimated that the standard deviation of the residence time ranged between ±11 s and ±3 min for mean residence times t of 20 s and 15 min, respectively. The rate of heat transfer is of crucial importance in order to study aggregation at short times scales. A simple heat balance on an elementary volume of the liquid makes it possible to relate the temperature gradient along the tube length to the global heat transfer coefficient h accounting for internal and external convection as well as heat conduction through the tube walls. The mean temperature T at abscissa x from the point where the tube was immersed is given by

T (x) -T ∞ T 0 -T ∞ = exp - x λ , (3)
where λ = ρc p Q πdh is the thermal entrance length and ρ and c p are the density and specific heat of the aqueous solution. By substituting Eq. 1 in Eq. 3 with x = L, T can be expressed as a function of the residence time t

T (t) -T ∞ T 0 -T ∞ = exp - t τ , (4)
where the thermal characteristic time τ is given by

τ = ρc p d 4h . (5)
In the range of Re investigated (Re 1000), the heat transfer coefficient is independent of the flow rate [START_REF] Incropera | Fundamentals of heat and mass transfer[END_REF], so that this characteristic time remains unchanged when the flow rate is varied. This simple heat transfer model was tested experimentally by measuring the temperature at the outlet of the tube for various flow rates and tube lengths, Fig. 2. The data superimposes remarkably on an exponential decay law, as predicted, with a characteristic time of 1.30 s for the 0.81 mm tube. This strongly indicates that the model is adequate to describe the thermal history and that the heat transfer coefficient is independent of the flow rate. From this characteristic time value, the heat transfer coefficient is about h 640 W/K/m 2 and is similar to the coefficient that could be estimated assuming that transfers were limited by heat conduction through the tube walls. Importantly, this value of the thermal characteristic time allowed us to approximate the exponential function by a step function for times longer than a few tens of seconds. The heating rate was slower for the 0.25 mm tube, as it was thicker. From the thermal resistance of this tube, we estimate a thermal characteristic time of 3.38 s. This tube was only used for experiments at shear rates and residence time of more than 2000 s -1 and 60 s, respectively.

The residence time was limited to 15 min as the quantification of the fraction of remaining native-like WPI at this time showed that 5% of native WPI were remaining. This quantification was achieved according to the loss of solubility of the denaturated protein at pH 4.6 [START_REF] Dannenberg | Reaction kinetics of the denaturation of whey proteins in milk[END_REF]. Non-native like proteins and aggregates were precipitated by pH adjustment and the solution was then centrifuged at 20.000 × g for 20 min. The concentration of remaining native-like WPI in the supernatant was determined by UV absorption at 280 nm.

Test tube aggregation process

Aggregation under static conditions was induced in a polypropylene "test tube" of 5 mL (Eppendorf tube). This aggregation process is referred to as test tube process later on in this article and was used as a control with shear rates limited by natural convection. In this setup, isothermal conditions were reached after 3 minutes.

Small-Angle X-Ray Scattering

SAXS experiments were performed at ESRF (Grenoble) on beamline ID02 which offered an energy of 12.46 keV and a wide range of wave vector q, from 3 × 10 -3 to 4 nm -1 . Samples were analyzed under static conditions in a glass capillary of 2 mm diameter. This method serves to measure the scattering of several samples within the same capillary, without handling it, and thus to accurately substract the contribution of the tube and the solvent from the scattering spectra. To minimize Xray damage of the sample, the exposition time was set at 0.5 s. For each experimental condition tested, the background of the cell containing water and 4 spectra were acquired. Spectra of aggregates were obtained by averaging the 4 data sets and subtracting the background. For each condition, the sample was diluted up to a factor 100 with demineralized water to ensure that the spectrum could be assimilated to the form factor of the suspension of aggregates, i.e. the scattered intensity was linear with respect to the WPI concentration, in the whole range of wave vector. Diluting the sample with a solution of equivalent pH and ionic strength had no effect on the spectra (data not shown).

In particular, we focused on two ranges of wave vectors. In the Guinier regime (q 1/R g ), the scattered intensity I of a dispersion of aggregates is given by

I(q) = I 0 1 + q 2 R 2 g /3 (6)
where R g is the radius of gyration of the aggregates and

I 0 = KC agg M w , ( 7
)
where K is a constant, C agg is the concentration aggregated proteins and M w is their mass. I 0 and R g were obtained by a least-square fitting of I as a function of 1/q 2 . Conformational changes in the structure of the proteins were detected in the high range of wave vector (0.5-3 nm -1 ) by plotting Iq 2 as a function q 2 (Kratky plot, [START_REF] Pérez | Heatinduced unfolding of neocarzinostatin, a small all-β protein investigated by small-angle x-ray scattering[END_REF], which allowed us to follow the kinetics of denaturation.

Resonant Mass Measurement

The mass of individual aggregates M w and its distribution were measured independently from SAXS analysis by resonant mass measurement (RMM, Archimedes, Malvern). A highly diluted dispersion was injected in a microfluidic channel with a micro-cantilever (Hi-Q Micro Sensor), for which the resonant frequency was calibrated. When a particle goes through the channel, the resonant frequency of the cantilever changes depending on the buoyant mass of the particle.

The buoyant mass (M B ) is proportional to the frequency shift ∆ f

M B = ∆ f S ( 8
)
where S is the sensitivity of the resonator. The conversion of M B into dry mass of aggregates M w is based on the density of whey protein ρ WPI (1.32 g/mL) and the density of water ρ w [START_REF] Weinbuch | Micro-flow imaging and resonant mass measurement (archimedes)-complementary methods to quantitatively differentiate protein particles and silicone oil droplets[END_REF] 10 -3 10 -2 10 -1 10 0 10 1 10 0 10 5

Continuous aggregation Test tube aggregation Guinier fit Guinier fit

Figure 3: SAXS intensities I(q) of the WPI aggregates obtained after 15 min of heat treatment at 92 • C in a test tube (dashed line) and our millifluidic continuous process (dashed-dotted line), γ = 53 s -1 , d = 0.81 mm, Q = 15 mL/h, L = 7.3 m. Guinier fits (solid lines) gave R g of 74 and 225 nm for test tube and continuous processes, respectively.)

M w = M B 1 -ρ w ρ WPI (9)
An archimedes device could detect particles with a mass of at least 4.5 × 10 -16 g. A few thousands of aggregates were analyzed per sample, which enabled us to determine the probability density function (PDF) of aggregates mass weighted by their mass.

Results and Discussion

Comparison with aggregation in test tube

The main advantage of our millifluidic process of aggregation was to heat the solution of proteins at 92 • C in few seconds and to control shear rates in laminar conditions (Figure 2). For a residence time of 15 min, we first compared the aggregates produced using the continuous millifluidic process (γ = 53 s -1 ) and those using a more standard process in a test tube of 5 mL. In this second case, shear rates were limited to some natural convection and isothermal conditions were reached after 3 min.

The aggregates showed large differences on length scales larger than 10 nm, whereas there was no difference at smaller length scales (Fig. 3). For q larger than 0.1 nm -1 , the scattered intensities superimposed, meaning that the small scale structures of the aggregates were similar for both conditions. Moreover, quantification of the fraction of native WPI by pH adjustment to 4.6 [START_REF] Delahaije | Controlling the ratio between native-like, nonnative-like, and aggregated β-lactoglobulin after heat treatment[END_REF] showed that the aggregation was almost complete at the end of the thermal treatment for both processes; the residual quantity of native WPI being only 5%. This statement was supported by the strong similarities of I in the q range of 0.5-2 nm -1 . However, large differences were evidenced at lower q in the Guinier regime with striking differences in terms of radius of gyration R g and scattered intensities I 0 (Eq. 6). R g in test tube was 74 nm, i.e. three times lower than in the case of the millifluidic process (225 nm). I 0 of aggregates obtained by millifluidic continuous process was 21 times bigger than the value of I 0 of aggregates obtained by test tube process. As C nat /C 0 was about 0.05 at 15 min, we concluded that aggregation is almost complete for both processes after 15 min. Thus, the variation on the value of I 0 between the two conditions reflected directly the variations of mass of the aggregates (I 0 ∼ M w , Eq. 7). We inferred their relative density,

ρ b /ρ c ∼ I b 0 /I c 0 R c g /R b g 3
, where ρ is the density in WPI of the aggregates and b and c refer to test tube and continuous conditions, respectively. We did not find any significant density differences between the two processes, ρ b /ρ c ∼ 1.3. At first sight, this result does not fit with the findings of [START_REF] Spiegel | Whey protein aggregation under shear conditions-effects of lactose and heating temperature on aggregate size and structure[END_REF] of an increasingly compact aggregate structure with temperature. However, the thermal history was characterized by a slow heating across the range of temperature in our test tube, whereas isothermal conditions were achieved in the tubular process used by [START_REF] Spiegel | Whey protein aggregation under shear conditions-effects of lactose and heating temperature on aggregate size and structure[END_REF].

Two main differences between the two processes could be put forward to explain these contrasts: the thermal history and the fact that the continuous process involved a flow. The temperature strongly affects the various stages of WPI aggregation, their kinetics, their size [START_REF] Spiegel | Whey protein aggregation under shear conditions-effects of lactose and heating temperature on aggregate size and structure[END_REF][START_REF] De Guibert | Flow process and heating conditions modulate the characteristics of whey protein aggregates[END_REF][START_REF] Petit | βlactoglobulin denaturation, aggregation, and fouling in a plate heat exchanger: pilot-scale experiments and dimensional analysis[END_REF] but also the protein conformations [START_REF] Pérez | Heatinduced unfolding of neocarzinostatin, a small all-β protein investigated by small-angle x-ray scattering[END_REF][START_REF] Tolkach | Reaction kinetic pathway of reversible and irreversible thermal denaturation of β-lactoglobulin[END_REF]. However, we did not observe significant differences in the high q range of the spectra between the two conditions, where protein conformation changes could be observed (see next section). In the continuous process, aggregation kinetics took place in a Poiseuille flow, which exhibited a gradient of velocity. Few studies showed that the shear rate could influence the size of large aggregates [START_REF] De Guibert | Flow process and heating conditions modulate the characteristics of whey protein aggregates[END_REF][START_REF] Wolz | Thermal aggregation of whey proteins under shear stress[END_REF][START_REF] Simmons | The effect of temperature and shear rate upon the aggregation of whey protein and its implications for milk fouling[END_REF]. The shear rate was also non-zero in the test tube due to natural convection, but should be smaller than the mean shear rate investigated in the small scale continuous process (53 s -1 ). At this stage, it seemed difficult to further interpret the above result without the additional experiments that are detailed hereafter.

Kinetics of aggregation in isothermal conditions

The reaction and aggregate formation kinetics was studied using the millifluidic continuous process. We studied these kinetics along a range of residence time t (20 s -15 min), for which the thermal history was approximated by a step function from 22 to 92 • C (Figure 2).

R g and M w evolution of WPI aggregates with t for fixed values of γ (32 and 53 s -1 ) were determined by SAXS and RMM, Figure 4. The results showed unambiguously that R g and M w remained almost constant from 20 s to 15 min of aggregation, with R g 205 nm and M w 10 -14 g. The slight increase of M w remained in the order of the measurement uncertainty. We were also aware that the continuous process tended to average the tendencies over time due to diffusion and hydrodynamic dispersion in the longitudinal direction of the flow (see Section 2.2). The probability density functions (PDF) of M w revealed that the suspension of aggregates had a polydispersity with a standard deviation of 30% of the mean value. Our results were in line with those of de Guibert et al. (2020) who studied WPI aggregation under similar physico-chemical parameters. The main differences were the thermal history and flow conditions, which were more complex (laminar and turbulent conditions) than those of this work. They also revealed the existence of a second population of WPI aggregates of smaller size by fieldflow fraction. However, RMM could not detect masses smaller than 4.5 × 10 -16 g. The aggregation number, which depicts the number of protein particles making up an aggregate, was about of N agg 3.3 × 10 5 . The fractal dimension being 2 at pH 7.0 [START_REF] Gimel | Structure and distribution of aggregates formed after heat-induced denaturation of globular proteins[END_REF][START_REF] Mehalebi | Light scattering study of heatdenatured globular protein aggregates[END_REF][START_REF] De Guibert | Flow process and heating conditions modulate the characteristics of whey protein aggregates[END_REF], we estimated the size of the aggregates, as R g = b N agg /6, with b the size of the β-lactoglobulin [START_REF] Moitzi | Structure of β-lactoglobulin microgels formed during heating as revealed by small-angle x-ray scattering and light scattering[END_REF], the main component of WPI. We found 220 nm, in very good agreement with the value estimated from the Guinier regime.

In contrast with the size, the scattered intensity at low q, I 0 , increased progressively with t. Since the aggregate mass was roughly constant, I 0 was directly proportional to the concentration of aggregates C agg (Eq. 7). Therefore, the aggregate concentration increased significantly from 20 to 915 s, by a factor of about 4.

This first set of results showed that the aggregate growth happened at short time scales (less than 20 s) and their size was limited, whereas WPI were continuously aggregated on time scales of few tens of minutes. These observations were consistent with previous studies (de [START_REF] De Guibert | Flow process and heating conditions modulate the characteristics of whey protein aggregates[END_REF][START_REF] Zuniga | Kinetics of formation and physicochemical characterization of thermally-induced βlactoglobulin aggregates[END_REF][START_REF] Roefs | A model for the denaturation and aggregation of β-lactoglobulin[END_REF], except that at lower temperature the aggregates grew at time scales of a few minutes.

The kinetics of proteins unfolding was studied by focusing on the high q range of SAXS spectra, which is sensitive to the protein conformation. The Kratky plots displayed in Fig. 5 exhibited a significant change as a function of t. For the native proteins (t = 0), a bell-shaped curve was seen, whose the peak position corresponded to the length scale of β-lactoglobulin. The peak intensity decreased when the residence time increased. For the longest residence time (915 s), this peak was very weak. This bell-shape curve is a signature of a globular protein [START_REF] Rambo | Characterizing flexible and intrinsically unstructured biological macromolecules by sas using the porod-debye law[END_REF][START_REF] Bernado | Structural analysis of intrinsically disordered proteins by small-angle x-ray scattering[END_REF], and the decrease of its intensity reflects the unfolding of the proteins upon heating [START_REF] Pérez | Heatinduced unfolding of neocarzinostatin, a small all-β protein investigated by small-angle x-ray scattering[END_REF]. Taking advantage of this feature and the fact that the form factor is a linear combination of the form factors of the different states of the protein in solution [START_REF] Pérez | Heatinduced unfolding of neocarzinostatin, a small all-β protein investigated by small-angle x-ray scattering[END_REF], i.e. native, partially unfolded and irreversibly aggregated, we estimated the progress of the unfolding process.

In the range 1-2 nm -1 , and as evidenced in Fig. 5, all the spectra were described by a linear combination of the spectrum of the native protein solutions (I nat , at t = 0) and of the final spectrum (I f , t = 915 s), i.e the presence of partially unfolded proteins was neglected,

I(t, q) = α(t)I nat (q) + [1 -α(t)] I agg (q) (10) with α(t) = C nat (t)/C 0 .
The acid precipitation test carried out on the final solution (at t = 915 s) revealed that it contained about 5% of native proteins. Thus, I f = 0.95I agg + 0.05I nat , where I agg is the intensity of a fully aggregated solution. The remaining fraction of native proteins was inferred from the value of α for a given aggregation time, as

C nat (t) C 0 = 0.95α(t) + 0.05. ( 11
)
The fraction of native proteins decreased in the first 60 s, a time at which half of the proteins were unfolded (Fig. 5). Note that the estimation of C nat by the test of the loss of solubility of the denaturated protein at pH 4.6 was coherent with the values obtained by SAXS.

This kinetics of unfolding was compared with that of aggregated WPI, C agg (t). Since the aggregate mass was approximately constant, the fraction of aggregated proteins was directly inferred from the time evolution of the low-q SAXS intensity I 0 :

C agg (t) C 0 = 0.95 I 0 (t) I 0 (t = 915s) . ( 12
)
Note that this quantification of C agg was independent of that of C nat , which was inferred from high-q SAXS intensity plot.

The symmetry between C agg /C 0 and C nat /C 0 is striking (Fig. 5). Moreover (C agg (t) + C nat (t))/C 0 varied by ±15% around 1 during the entire aggregation process. This clearly shows that the rate of protein unfolding was that of protein aggregation. This result highlights the self-consistency of our approach and confirms that WPI aggregation occurs as soon as a nucleus is present. it is worth mentioning that the above analysis in the high-q range assumes only two states for the protein conformation, which was an oversimplified picture. However, the fact that (C agg (t) + C nat (t)) C 0 justifies a posteriori this assumption.

The order of the reaction n is defined as dC agg /dt = kC n nat , where k is the reaction rate constant, which is supposed to be a function of temperature and solvent composition. Reaction orders n were reported to vary between 1 and 2 [START_REF] Dannenberg | Reaction kinetics of the denaturation of whey proteins in milk[END_REF][START_REF] De Wit | Thermal stability and functionality of whey proteins[END_REF][START_REF] Erabit | Coupling population balance model and residence time distribution for pilot-scale modelling of β-lactoglobulin aggregation process[END_REF][START_REF] Wolz | Thermal aggregation of whey proteins under shear stress[END_REF][START_REF] Roefs | A model for the denaturation and aggregation of β-lactoglobulin[END_REF][START_REF] Phan-Xuan | On the crucial importance of the ph for the formation and selfstabilization of protein microgels and strands[END_REF][START_REF] Zuniga | Kinetics of formation and physicochemical characterization of thermally-induced βlactoglobulin aggregates[END_REF]. From kinetic models of β-lactoglobulin that recognized that the rate of nucleation (initiation) is much smaller than the rate of propagation (aggregation), we expected a n of 1.5 [START_REF] Roefs | A model for the denaturation and aggregation of β-lactoglobulin[END_REF][START_REF] Tolkach | Reaction kinetic pathway of reversible and irreversible thermal denaturation of β-lactoglobulin[END_REF]. n = 1.5 was also predicted theoretically for rapid aggregation theory of colloidal solutions [START_REF] Smoluchowski | Versuch einer mathematischen theorie der koagulationskinetik kolloider lösungen[END_REF]. Our results were consistent with an order of 1.5 (subpanel in Fig. 5-b) and confirm that nucleation is the limiting step.

A simplified reaction scheme for the aggregation may consist in two main steps, the first one is a nucleation step, where one or a few (n 0 ) native proteins N undergo conformational change and irreversible association to form a precursor P for subsequent aggregation in aggregates A i made of i proteins

n 0 N → P P + N → A n 0 +1 A n 0 +i + N → A n 0 +i+1
This scheme is that of a standard nucleation and growth process. It does not detail the initial conformational changes leading to the first irreversible precursor P, but since these are reversible with a very short charasteristic time [START_REF] Roberts | Non-native protein aggregation kinetics[END_REF], they have limited impact on the aggregation scenario except concerning the dependency with initial concentration C 0 , which is not studied here. It also ignores possible combinations between aggregates. To fully model the above results, this reaction scheme should involve a kind of termination step in order to limit the growth of the aggregates to a given size [START_REF] Roefs | A model for the denaturation and aggregation of β-lactoglobulin[END_REF][START_REF] Roberts | Non-native protein aggregation kinetics[END_REF]. The possible mechanisms are discussed hereafter. . b : the fraction of native WP, C nat /C 0 was determined from high q-range of spectra (•, Eq. 10, 11) and was calculated from a linear combination of the spectra of natives and aggregated WPI in the q-range 1-2 nm -1 (dashed lines), see text for details. The fraction of aggregated WP, C agg /C 0 , was determined independently from the low q-range of spectra ( , Eq. 12). Dashed lines are guide for the eyes. In insert, kinetic plot (C nat /C 0 ) 1-n versus t with reaction order n= 1.5. γ = 53 s -1 , d = 0.81 mm, Q = 15 mL/h, L = 10 -87 cm

The above detailed experimental results might be summarized as follows: (i) Denaturation and aggregation occured consequently, since intermediate states of WPI (i.e. unfolded but not aggreagted) were negligible at all times, (ii) Aggregates were formed within a few seconds, much faster than the global reaction rate, and their concentration kept growing over time with constant size and mass. The second one implied two features: on the one hand aggregation is not the limiting step and on the other hand there exists an additional mechanism which limits the growth of the aggregates when they have reached a given size. Other scenarios are not compatible with these two key observations. In particular, if aggregation was the limiting step, then the aggregate size would have increased significantly during the process.

We could only speculate on the scenario which limits the growth of the aggregates to a given size (for the studied experimental conditions, about 200 nm). Based on analogy with radical polymerisation chemistry, [START_REF] Roefs | A model for the denaturation and aggregation of β-lactoglobulin[END_REF] suggested that the polymerization process of β-lactoglobulin stopped when one reactive intermediate A n 0 +i reacted with another one forming an aggregate without a reactive sulhydryl group. Under this assumption, they predicted that M w ∼ C 0.5 0 . Their reaction scheme was consistent with their kinetics of aggregation at low temperature (65 • C). However, [START_REF] Zuniga | Kinetics of formation and physicochemical characterization of thermally-induced βlactoglobulin aggregates[END_REF] moderated this concept, as their experiments at 80 • C insinuated that small molecular weight intermediates were cross-linked by disulfide bonds and formed larger aggregates via noncovalent bonding (e.g. by hydrophobic interactions). Competition between short-range depletion attraction and long-ranged electrostatic repulsion also leads to equilibrium clusters of finite-size whose number of aggregation N agg is proportional to C 0 [START_REF] Stradner | Equilibrium cluster formation in concentrated protein solutions and colloids[END_REF][START_REF] Campbell | Dynamical arrest in attractive colloids: The effect of long-range repulsion[END_REF][START_REF] Cardinaux | Modeling equilibrium clusters in lysozyme solutions[END_REF]. However, some of our preliminary results would invalidate the above-mentioned mechanisms as we found that R g was con-stant for C 0 ranging from 0.4 to 4% w/w at 92 • C and residence time t of 60 s, in agreement with [START_REF] Mehalebi | Light scattering study of heatdenatured globular protein aggregates[END_REF]; Phan-Xuan et al. ( 2011)(pH 7.0, no added salt, 85 • C and C 0 ≤ 7 g/L). Alternatively, our results could be explained by the mechanism introduced by Phan-Xuan et al. (2011) in which a selfstabilization effect could appear due to spontaneous increase of the pH during aggregate formation that should lead to an increase of their surface charge density and inhibits secondary aggregation. Similarly to the association of charged amphiphilic polymers [START_REF] Zhulina | Theory of block polymer micelles: Recent advances and current challenges[END_REF], counterions should be partly repelled from the aggregates and may lead to this electrostatic repulsion. At this stage, we have no experimental evidence that could validate or not one of these mechanisms. Additional studies varying the ionic force and the concentration of WPI would be interesting to test these various hypotheses.

Effect of shear rate

The flow rate Q was one of the important control parameters of the millifluidic continuous process that allowed us to increase the production rate, but also the mean shear rate γ (Eq. 2) in laminar conditions. The general picture depicted by previous studies is that shear rate could be a control parameter of the aggregate size. For low and moderate shear rate, the size should increase as the rate of collision between monomers and oligomers increases. At higher shear rates, the size should decrease as shear forces could break down the aggregates. An optimum could exist [START_REF] Simmons | The effect of temperature and shear rate upon the aggregation of whey protein and its implications for milk fouling[END_REF][START_REF] Wolz | Thermal aggregation of whey proteins under shear stress[END_REF]. However, this picture is not clearly established for thermal denaturation of proteins due to the difficulty of uncoupling thermal history from flow conditions [START_REF] Erabit | Coupling population balance model and residence time distribution for pilot-scale modelling of β-lactoglobulin aggregation process[END_REF][START_REF] De Guibert | Flow process and heating conditions modulate the characteristics of whey protein aggregates[END_REF]. We studied a range of shear rates γ which extended from 50 s -1 to 2.66×10 3 s -1 (Fig. 6). The comparison was made for an intermediate residence time t of 60 s for which one half q (nm -1 ) (b)

1/q 2 (nm 2 ) of native WPI were aggregated and so Kratky plots were sufficiently distinct from native and fully aggregated proteins. This particular choice had no major consequences since we reported that the structural properties of the aggregates (mass and size) did not significantly depend on the residence time.

The Kratky plots for the three shear rates investigated superimpose in the high-q range (Fig. 6-a). This feature showed unambiguously that the aggregation kinetics was not changed when γ was increased. For very high shear rates, the flow could influence the frequency of the collisions between reactants and thus increase the reaction rate. The relevant Péclet number for this effect is Pe = γl 2 /D, where l is a characteristic distance between the considered reactants, which was of the order of 10 nm for the proteins, but significantly higher for the aggregates. For the highest shear rate tested, this Péclet number remained lower than unity (Pe 0.2), meaning that diffusion remained the dominant mechanism. It is thus not surprising that the shear rate did not influence the aggregation rate, provided that the nucleation stage is the limiting one.

In the Guinier regime, the data plots in Fig. 6-b exhibit a small trend. Both I 0 and R g slightly increased when the shear rate was increased. R g increased by about 10% for variations of γ of almost two orders of magnitude. From the values of I 0 obtained by the best Guinier fits and the value of C agg = 1 -C nat inferred by a linear combination in the Kratky plots (see previous section and Fig. 5), we calculated the aggregate mass M w ∼ I 0 /C agg (Eq. 7). M w increased slightly with γ, similarly to the radius of gyration. We were aware that these variations are of the order of magnitude of the measurement uncertainties, whereas γ varied by almost two orders of magnitude. We concluded that the shear rate had no significant influence on the aggregation or on the denaturation in isothermal conditions.

We returned our attention to the marked size differences between aggregates prepared using the millifluidc continuous process and the ones prepared in test tube (Fig. 3). We inferred that the shear rate was not responsible for this large increase of the aggregate size when switching from test tube to continuous process for the physico-chemical conditions investigated. The difference in heating rate was more likely to be responsible for the difference between test tube and continuous process. As in most processes, the scales are of a few millimeters, it is impossible to separate thermal from flow conditions. In light of the above results, our opinion was that some effects attributed to the influence of shear rate in previous studies [START_REF] Simmons | The effect of temperature and shear rate upon the aggregation of whey protein and its implications for milk fouling[END_REF][START_REF] Wolz | Thermal aggregation of whey proteins under shear stress[END_REF] could be in fact attributed to large differences of thermal history. Additional studies varying the thermal history could be of interest to confirm that the latter could be used to tune the size of the aggregates.

Conclusions

In this paper, we studied the denaturation and aggregation of whey protein isolates under flow and at high temperature (92 • C) for a single physicochemical condition leading to submicrometric aggregates (neutral pH, 10 mM NaCl,WPI 4 % w/w) with a well-controlled thermal history. We showed unambiguously that the size and mass of aggregates reached a steady state value within a few seconds, whereas the time scale associated with the kinetics of aggregation was of a few minutes. Moreover, we calculated from the Kratky plots the kinetics of consumption of native proteins, which was symmetrical to the kinetics of aggregated proteins. It means that partially unfolded proteins were virtually absent and WPI aggregated as soon as they are unfolded. The order of the reaction was consistent with 1.5, typical for rapid aggregation of proteins [START_REF] Roefs | A model for the denaturation and aggregation of β-lactoglobulin[END_REF][START_REF] Tolkach | Reaction kinetic pathway of reversible and irreversible thermal denaturation of β-lactoglobulin[END_REF]. All these results lead us to conclude that the aggregation at 92 • C is limited by the nucleation and there is a mechanism that limits the size of the aggregates. The underlying mechanism is an open issue [START_REF] Roefs | A model for the denaturation and aggregation of β-lactoglobulin[END_REF][START_REF] Zuniga | Kinetics of formation and physicochemical characterization of thermally-induced βlactoglobulin aggregates[END_REF][START_REF] Stradner | Equilibrium cluster formation in concentrated protein solutions and colloids[END_REF][START_REF] Phan-Xuan | On the crucial importance of the ph for the formation and selfstabilization of protein microgels and strands[END_REF]. Different scenarios could be tested by varying protein concentration and / or ionic strength.

Moreover, we observed that the size of the aggregate and the denaturation is almost invariant with the shear rate (for variations covering almost two orders of magnitude) in isothermal conditions. We concluded that shear rate does not influence aggregation for the conditions tested and thermal history can be reasonably considered as responsible for the huge difference in terms of aggregate size between our millifluidic process and test tube conditions. This latter finding highlights the interest of studying growth of aggregates and unfolding with various thermal kinetics. The approach that we used in this work makes it possible to simultaneously characterize denaturation, nucleation and aggregation. We think it could be of great interest to extend it to other temperatures, especially higher ones which have been described as aggregation limited conditions [START_REF] Dannenberg | Reaction kinetics of the denaturation of whey proteins in milk[END_REF].

Figure 1 :

 1 Figure 1: Diagram of the millifluidic continuous aggregation process. WPI solution flowed in 0.81 mm internal diameter tube at a flow rate Q and a temperature T 0 . The tube was immersed in a heated water bath at temperature T ∞ . From the time the tube was immersed (x = 0), the fluid temperature quickly rose along a characteristic length λ. The mean residence time t is directly proportional to the length L of the tube.

Figure 4 :

 4 Figure 4: Kinetics of WPI aggregation. a: Guinier plot of SAXS intensity of aggregated WPI solutions for various residence times t at γ = 53 s -1 Solid lines are the fit of Guinier equation, Eq. 6. d = 0.81 mm, Q = 15 mL/h, L = 16.2 -740 cm. b: Probability density function (PDF) of the mass of aggregates M w for various t at γ = 32 s -1 as determined by RMM. d = 0.81 mm, Q = 9 mL/h, L = 10 -87 cm. c: Mean value of M w as a function of t, as determined by RMM. d: Radii of gyration R g , which were obtained by the Guinier fit, as a function of t. Error bars represent ±10% of the mean value.

Figure 5

 5 Figure5: a: Kratky plots showing the change of WP conformation with the residence time t. . b : the fraction of native WP, C nat /C 0 was determined from high q-range of spectra (•, Eq. 10, 11) and was calculated from a linear combination of the spectra of natives and aggregated WPI in the q-range 1-2 nm -1 (dashed lines), see text for details. The fraction of aggregated WP, C agg /C 0 , was determined independently from the low q-range of spectra ( , Eq. 12). Dashed lines are guide for the eyes. In insert, kinetic plot (C nat /C 0 ) 1-n versus t with reaction order n= 1.5. γ = 53 s -1 , d = 0.81 mm, Q = 15 mL/h, L = 10 -87 cm

Figure 6 :

 6 Figure6: Effect of shear rate on denaturation and aggregation. a: Kratky plot showing the invariance of WP conformation for the different shear rates investigated for t = 60 s . b: Guinier plot of SAXS intensity for various shear rates at t = 60s. Solid lines are the fits using the Guinier equation (Eq. 6). c: Radii of gyration R g as a function of γ. d: I 0 /C agg as a function of γ. C agg = 1 -C nat was determined from the Kratky plot. d = 0.81 mm and 0.25 mm for γ = 2.5 × 10 3 s -1 , Q = 15 -150 mL/h, L = 48.6 -750 cm.

Acknowledgements

The authors thank gratefully Theyencheri Narayanan (ESRF), Michael Sztucki (ESRF), William Chèvremont (LRP), Antoine Naillon (LRP), Mehdi Maleki (LRP) and Luis Carlos Morocho Rosero (LRP) for their technical support on the ID02 beamline of ESRF. We thanks also Frédéric Pignon (LRP) and Javier Perez (SOLEIL) for insingthfull discussions. LRP is part of the LabEx Tec 21 (Investissements d'Avenir -grant agreement ANR-11-LABX-0030) and Institut Carnot PolyNat (Investissements d'Avenir -grant agreement ANR-11-CARN-030-01).