P Švančara

P.-E Roche

M La Mantia

Spatial distribution of the quantized vortices' tangle in thermally driven round jets of superfluid helium

Thermally driven flows of superfluid 4 He display unique features, often related to the presence of quantized vortices -line singularities embedded in the liquid. Here we focus on turbulent round jets, experimentally investigated using the flow visualization and second sound attenuation techniques, at Reynolds numbers exceeding 10 4 . These turbulent flows are driven by releasing heat into a small volume of liquid, open to the surrounding bath through a cylindrical nozzle, 2 mm in diameter. Our measurements reveal in unprecedented detail how the tangle of quantized vortices associated to the jets arranges itself in space, for distances ranging from 9 to 34 nozzle diameters, at fluid temperatures between 1.64 and 2.10 K. We specifically find that the vortex tangle spreads in the radial direction, while it dilutes away from the nozzle. Additionally, the tangle density is found to systematically depend on the flow forcing. Two physical interpretations of the observed behavior are proposed, which could motivate further investigations of this peculiar flow. One leads us to conjecture a self-similar functional form of the vortex tangle density across counterflow jets. The other suggests that the position of the superfluid stagnation point -a characteristic feature of counterflow jets -could depend on the flow forcing as well.

I. INTRODUCTION

Turbulent jets are typical representatives of boundary-free shear flows that can be realized not only in classical Newtonian fluids but also in superfluid 4 He -see e.g. Refs. WF69 and DSD + 11. A jet flow is usually obtained by driving a fluid stream through a nozzle into a large reservoir. The interaction between the injection flow, the nozzle and the fluid inside the reservoir results in a distinct, spatially inhomogeneous structure that can be experimentally studied. In the present work we focus on turbulent round jets submerged in superfluid 4 He, also known as He II. This phase of liquid 4 He exists only at temperatures below approximately 2.2 K, at the saturated vapor pressure, and its behavior is appreciably affected by the quantum order that spontaneously develops in the fluid. Macroscopic consequences of this order are, for example, the possible occurrence of inviscid flows and the coupling between temperature and pressure gradients (the fountain effect) -see Refs. BSS14 and MJS18 for reviews on this unique liquid.

At temperatures higher than about 1 K thermal excitations cannot be neglected and the large-scale hydrodynamics of He II is usually described by the phenomenological twofluid model. Lan41,Don09 The gas of excitations can be seen as a viscous fluid called the normal component, while the quantum-ordered system behaves as an inviscid, superfluid component. The two fluid components are fully miscible and their density fractions, which depend only on temperature, are plotted in Fig. 1 for the range of temperatures relevant here -note in passing that the He II density ρ ≈ 145 kg/m 3 depends weakly on temperature. DB98 FIG. 1. Left axis: temperature dependence of the density fractions of the normal (pn, dot-dashed line) and superfluid (ps, dashed line) components of He II. Right axis: temperature dependence of the second sound speed (solid line). T λ ≈ 2.2 K denotes the superfluid transition at the saturated vapor pressure. Data from Ref. DB98.

Additionally, the circulation of the superfluid component is quantized. Don91 Specifically, each circulation quantum κ ≈ 10 -7 m 2 /s is associated to a quantized vortex, which is a line singularity of the quantum order parameter characterized by a narrow core (approx. 0.1 nm) and macroscopic length. When superfluid 4 He becomes turbulent, quantized vortices tend to interact and eventually form a dynamic tangle, which is the key component of turbulence in He II. BSS14,MJS18 Here we employ experimental tools to study the quantized vortices' tangle associated to steady turbulent jets, in view of understanding in what aspects jet flows in superfluid 4 He may differ form their classical analogs. The quantized vortices' tangle plays an important role in this context because it provides the coupling mechanism between the otherwise independent components of He II. At flow scales larger than the mean distance between quantized vortices the magnitude of this interaction -usually called mutual friction force -is proportional to the difference between the velocities of the normal (v n ) and superfluid (v s ) components. BSS14,MJS18 According to the just outlined model, when heat is dissipated in He II, two velocity fields pointing in opposite directions are established, resulting in the so-called thermal counterflow. The normal component flows away from the heat source, dispersing heat into the 4 He bath, while the superfluid component, which has zero entropy and therefore cannot transport heat, flows in the opposite direction, to compensate for the mass flow of the normal component. In the widely studied case of channel counterflow, a heating power P is dissipated at the closed end of a channel with cross-section S, submerged in a liquid bath kept at temperature T . In the steady state the normal component moves towards the open end of the channel with the average velocity

v n = q ρsT , (1)
where q = P/S is the dissipated heat flux, ρ indicates the density of He II and s is the fluid specific entropy. In order to satisfy the null mass flow rate condition just mentioned, the superfluid component flows towards the heater with the average velocity v s in such a way that p n v n = -p s v s holds (we assume here that v n > 0 and v s < 0). As a result, the relative velocity of the two components is in general nonzero. We usually denote this velocity difference as the counterflow velocity v ns , which is then given as

v ns = v n -v s = q ρ s sT = v n 1 -p n , (2)
where ρ s = p s ρ is the density of the superfluid component.

Here we take advantage of the just outlined flow regime because our turbulent jets are driven thermally, by dissipating a constant heating power in a small enclosed volume open to the He II bath via a cylindrical nozzle. The resulting flow is usually named thermal counterflow jet -see Ref. LL84 for a review on early works -and in the past it was mainly regarded as a playground system for the study of the mutual friction force in the absence of solid boundaries, which significantly influence the development of turbulence in He II, as recently discussed in Refs. BLR17, LM17, HvLM18, and vHRLM18.

Following Ref. MI89 we define a Reynolds number relevant for thermal counterflow jets as

Re = ρv n d µ n , (3)
where d indicates the nozzle diameter and µ n denotes the dynamic viscosity of He II. As detailed below, the Re values achieved in the present work are larger than the critical value 5 × 10 3 reported in Ref. MI89, on the basis of flow visualization experiments. It then follows that our thermally driven jets are turbulent. Additionally, one expects that the mutual friction force is significantly enhanced in the nozzle output region because a dense tangle of quantized vortices develops within the nozzle itself, partly due to the interaction of the flow with the inner wall of the nozzle, and partly due to the mutual interaction between the normal and superfluid components. LM17 Consequently, the fast-moving normal component exits the nozzle and interacts with the vortices, which are flushed into the open volume, where their presence can be probed in various ways, e.g. by negative ion trapping. CCD68 Thermometric measurements carried out along the jet axis -see e.g. Refs. DB73 and MYN89 -reveal that a temperature difference ∆T develops inside the nozzle, i.e. the fluid temperature in the bath is smaller than that inside the nozzle. For example, the ∆T ∝ q 3 scaling obtained at heat fluxes higher than about 5 kW/m 2 -and typical of turbulent counterflow in channels GM49 -confirms that a dense tangle of quantized vortices should be present inside the nozzle in these conditions. However, the temperature gradient seems to disappear about one diameter away from the nozzle, as reported e.g. in Ref. DB73, at least for relatively high heat flux values, i.e. within the just mentioned cubic regime. On the basis of this evidence it was conjectured that the superfluid component might be coupled to the normal one in the free jet flow, resulting then in a nearly zero counterflow velocity (i.e. coflow of the normal and superfluid components) in the far field. The hypothesis was later supported by acoustic attenuation, Lag75 velocity-sensitive phase DL77 and laser Doppler velocimetry measurements, as close as 0.5d above the nozzle. NM94,NMK94 Additionally, it was found that the fluid velocity at the nozzle exit is apparently equal to p n v n , which can be explained by the transfer of momentum between the normal component, which exits the nozzle with a velocity v n , and the surrounding superfluid one, via a process similar to the turbulent entrainment known to occur for classical turbulent jets. LL84,LC03 It then follows that the flow field of the superfluid component must have a stagnation point near the nozzle. LL84,NMK94 We schematically sketch this velocity field in Fig. 2.

Additionally, following Ref. BFP12 one can split the flow field of turbulent jets into three regions. The near field region, extending up to 7 nozzle diameters away from the nozzle, can be considered as the region of flow establishment. Then, some self-similar properties of turbulent jets are expected to emerge in the intermediate field region and the flow is said to be fully developed only in the far field region, corresponding to more than 70 nozzle diameters away from the nozzle. Here, we focus on the intermediate field region of free counterflow jets, where the resulting single-component flow field of He II already resembles to some extent the Newtonian one, as shown, for example, in Ref. MTMN08. Specifically, it was reported that the radial profile of the flow velocity is nearly Gaussian BFP12,MTMN08 and that the profile width grows linearly with axial distance, the latter being associated to the similarity of the flow field. BV22,ZTBV22 The vertical component of the fluid velocity can then be written as in the cylindrical coordinates (r, φ, z), where v 0 is the centerline velocity, along the jet axis, and β indicates the jet growth rate. The centerline velocity was found to be constant for the first one to four nozzle diameters away from the nozzle MTMN08 and, for larger distances, the classical v 0 ∝ z -1 scaling holds. BV22,ZTBV22 Note also that in Ref. NMK94 the flow development region of counterflow jets was reported to be somehow shorter than that of ordinary turbulent jets.

v(r, z) = v 0 (z) exp - r βz 2 (4)
On the other hand, there is little known about the spatial structure of the quantized vortices' tangle in a thermal counterflow jet. In this work we aim to fill in this knowledge gap and we present the first spatially-resolved measurements of the vortex line density L(r, z) in this kind of flow, where L is defined as the total length of quantized vortices per unit volume. We specifically probe the quantized vortices' tangle using the second sound attenuation technique. Don09,VJSS19,WVR23 The latter is based on detecting temperature waves, which can be observed in flows of superfluid 4 He. Within the two-fluid framework, second sound waves consist of anti-phase oscillations of the normal and superfluid components, while the well-known density (pressure) waves -named first sound waves in He II -correspond instead to in-phase oscillations of the components. The speed of second sound is temperature dependent -it reaches a flat maximum near 1.65 K and steeply decreases near the superfluid transition temperature, as shown in Fig. 1 (right axis).

In summary, as detailed below, we find that the quantized vortices' tangle spreads in the radial direction, along the jet axis, in a way similar to that observed for the fluid velocity, while its peak density decreases away from the flow source. We then discuss, on the basis of several assumptions, the spatial distribution and the flow forcing dependence of the tangle density, directly obtained from the experimental data.

II. METHODS

The counterflow jets here investigated are obtained at the bottom of a standard 4 He cryostat by supplying constant heating power to a resistive heater located inside a purposemade enclosure, 3D-printed from copper filled PLA -see Ref. Š21 for a detailed description of the apparatus and for preliminary results that are further discussed in the following. The enclosure is terminated by a circular nozzle, with inner diameter d = 2 mm, as sketched in Fig. 3 -the nozzle is machined from brass and is 5d high. The thermally driven jets develop in the open volume above the nozzle -the diameter of this volume is approximately 75d and it is at least 350d high.

The vortex line density embedded in the flow is measured with a miniature second sound sensor RDD + 07,WVR21,WVR23 consisting of two silicon plates placed δ = (2.505 ± 0.002) mm apart. The plates support a heater-thermometer pair, microfabricated at their respective extremities. The sensor's cross-section in the direction of the flow at the measurement location is approximately twice 1 mm × 20 µm, and about 14 mm downstream this measurement location, the sensor's cross-section enlarges to approximately 2.5 mm × 3.5 mm. The heater is supplied with a sine-wave voltage signal with frequency f and, in consequence, second sound waves with frequency 2f are emitted towards the thermometer, which in turn develops weak periodic oscillations of its resistance. To read out this signal, the thermometer is supplied with a small constant current and we collect the corresponding voltage oscillations by a lock-in amplifier.

The two plates of the sensor form an open resonance cavity and standing second sound waves are obtained between them when the driving frequency f equals the resonance frequency f n , which can be approximated as

f n = nc 2 4δ , ( 5
)
where n is a positive integer and c 2 indicates the second sound speed, whose temperature dependence is plotted in Fig. 1 (right axis). By driving the device across resonance in quiescent He II, i.e. when the vortex line density is relatively small, we obtain a nearly Lorentzian resonance curve typical of harmonic oscillators, with amplitude A 0 and halfwidth ∆ 0 . In case the sensor is placed in a flow of He II containing a substantial vortex tangle, the second sound waves are attenuated by the action of the mutual friction force and the measured resonance amplitude A falls behind A 0 . The ratio A 0 /A > 1 can then be related to the vortex line density L. For a homogeneous and isotropic tangle, the latter can be approximately estimated VJSS19,WVR23 as

L = 6π∆ 0 Bκ A 0 A -1 , (6)
where B denotes the mutual friction parameter, tabulated in Ref. DB98. For a polarized (preferentially oriented) tangle, Eq. ( 6) is modified by a constant prefactor, VJSS19 i.e. the relation can be used as a qualitative estimate of L also for polarized tangles, assuming that the polarization does not significantly change over the course of the experiment. Additionally, in Ref. VJSS19 the uncertainty associated to Eq. ( 6) is reported to be of the order of 10 %. Still, we find its use in this study adequate, because such a level of uncertainty is comparable to the statistical uncertainty of A and to other systematic errors specific to the employed sensor, which are discussed below. Note also that the sensitivity of the second sound attenuation technique is further limited by the fact that it is based on the relative measurement of two second sound amplitudes. In practice, solely amplitudes meeting A/A 0 < ∼ 0.99 can be distinguished. For the typical value ∆ 0 = 300 Hz, we estimate from Eq. ( 6) that, at 1.95 K, the minimum density that can be faithfully detected is L min ≈ 6 × 10 8 m -2 . The latter corresponds to approx. 1.5 meters of vortex lines localized between the heater-thermometer pair, i.e. to a mean distance between quantized vortices of approximately 40 µm -the volume where L is measured is equal to approx. 2.5 mm 3 . The crucial feature of our sensor is its ability to be displaced within the experimental volume, thanks to a mobile shaft (see Fig. 3), without significantly altering the flow, due to its relatively small size and cross-section along the axial direction. Two room-temperature stepper motors control the shaft motion; one motor moves the shaft in the vertical direction and the other rotates it about its axis. In the latter case, the sensor moves inside the cryostat along a circle intentionally crossing the vertical axis of the setup. The angular deflection of the shaft α can then be used to obtain the radial coordinate r relative to the jet axis as

r = 2R sin α -α 0 2 , (7)
where R = 73 mm = 36.5d is the length of the horizontal mounting arm (see Fig. 3) and α 0 = 1.9 • ± 0.3 • denotes the offset between the zero position of the sensor, manually set before the experimental campaign, and its actual value, determined from the acquired second sound signal (see Fig. 4) -this radial coordinate is signed because the sensor is allowed to move on both sides of the jet axis, i.e. negative values of r simply denote that the corresponding deflection angle α < α 0 (and vice versa). Additionally, the sensor is mounted asymmetrically on the horizontal arm and the sensitive areas are located approx. 7d away from the support (see Fig. 3). As we show below, no significant differences in the acquired second sound signal are observed between strokes on opposite side of the jet's axis, which leads us to believe that possible blockage effects caused by the asymmetry in the mounting are negligible. Note also that possible blockage effects caused by the symmetrical body of the sensor have presumably the same strength, since both parts have comparable sizes and positions in the flow, i.e. possible blockage effects from the sensor body are expected to be negligible too. Finally, the vertical position z of the sensor, relative to the top of the nozzle, is obtained directly from the corresponding stepper motor and it can be determined with the accuracy of approx. 1 mm.

To the best of our knowledge, we report here the first use of a mobile second sound sensor to probe a spatially inhomogeneous tangle of quantized vortices. So far, inhomogeneous tangles were investigated by several static sensors placed along a counterflow channel -see Ref. VJSS19 for a few examples -or by multiple sensors located in different loci of the experimental setup, as e.g. in Ref. WVR21.

Before proceeding, we now mention two effects that might potentially bias the second sound measurements. First, in regions far away from the jet axis, the mean flow encounters the two plates of our sensor at nonzero angles of attack. The mean flow velocity at the location of the sensor, which can be up to approximately 0.1 m/s, as shown e.g. in Fig. 5, is significantly smaller than the velocities of the first and second sound waves -in the investigated temperature range these velocities reach their minima at 2.10 K, and are equal to 222 m/s and 13 m/s, respectively. DB98 Thus, no significant supersonic effects are expected to occur when the mean flow changes its direction due to the sensor. Nevertheless, a velocity boundary layer is expected to develop along each plate SPB17,LM17 and quantum vortices localized within these layers may contribute to additional attenuation of the measured signal. Specifically, boundary layer effects were investigated in turbulent flows of He II for similar sensors, at zero mean angle of attack. RDD + 07,WVR23 Their contribution was found to be negligible, partly due to the small thickness of the sensor plates, and partly due to the fact that the velocity boundary layer is expected to be small compared to δ (see Ref. WVR23 for a more detailed discussion of this effect). Additionally, in comparison to Ref. WVR21, the effect of the boundary layer on the measured signal might be larger, due to the nonzero angles of attack, but, at the same time, the distance between our sensor plates δ is 2.5 times larger than in the just cited paper, where δ ≈ 1 mm. In consequence, the contribution of boundary layers to the total signal attenuation is, in our case, reduced by a factor of roughly 2.5, at the cost of a lower spatial resolution.

The second source of bias is due to the advection of second sound waves from the resonance cavity, which also accounts for increased signal attenuation. This advection effect is present in all types of second sound resonators to a greater or lesser degree. WVR23 Specifically, the attenuation attributed to the vortex tangle is more significant and Eq. ( 6) represents an accurate estimate of L when the second sound sensor is operated with a low-order resonant mode. In our case, as detailed below, we use the 18 th harmonic mode, which means that the second sound beam is more focused and sensitive to advection by the mean flow than it would be for lower order modes. From the comparison with other available sensor geometries, we estimate this bias to be as large as a few tens of percent (a more detailed discussion of this effect is reported in Ref. WVR23). It is therefore more significant than the other bias mentioned above and, consequently, the additional sensitivity of the employed second sound sensor to the mean flow velocity should be kept in mind.

Aware of these drawbacks, we consolidate our second sound measurements using cryogenic flow visualization. The technique -discussed at length in Ref. Š21 -is based on following the flow-induced motions of micrometer-sized particles by a digital camera. We specifically visualize solid deuterium particles, which are illuminated by a 1 mm thick and 10 mm high laser sheet crossing the vertical axis of the jet. The acquired camera images are then processed to obtain particle trajectories and velocities. These measurements are carried out in an optical cryostat, with a 12 mm (6d) wide and 8 mm (4d) high field of view located approx. 10 mm (5d) above the the nozzle, along the jet axis -see again Ref. Š21 for experimental details.

III. EXPERIMENTAL RESULTS

Steady counterflow jets are investigated at three temperatures (approx. 1.65, 1.95 and 2.10 K) corresponding to normal component density fractions p n of 0.19, 0.48 and 0.74, respectively. Note that the helium bath was kept at constant temperature by maintaining the associated helium vapors at constant pressure, i.e. the temperature standard deviation was less than 12 mK at 1.65 K, and around 1 mK at higher temperatures. Depending on the employed experimental protocol, we split the acquired data sets into four categories, labeled A, B, C, and D -see Table I for relevant experimental conditions.

Data sets A1-A3 were collected near 1.65 K because the temperature dependence of the second sound speed has a plateau here, see Fig. 1 (right axis). The resonance condition given by Eq. ( 5) then becomes insensitive to small temperature fluctuations experienced in almost every cryogenic setup. Therefore, we continuously excite the sensor at the fixed frequency f = 37.22 kHz, which corresponds to the 18 th harmonic mode. By displacing the sensor, we sweep the volume above the nozzle and we then collect the second sound amplitude A for a set of positions given by (α, z). In order to increase the signal-to-noise ratio, multiple (at least 30) sweeps across the jet axis are performed for constant z and the resulting second sound amplitude is calculated as their average. The vortex-free resonance amplitude A 0 and the peak width ∆ 0 are obtained by fitting the full resonance curves, purposely acquired far away from the jet axis, i.e. far away from the tangle.

An example of the acquired second sound signals is presented in the top panel of Fig. 4. Here we show two sets of sweeps obtained near the extremities of the accessed axial distance, which are 9.0d and 31.5d, respectively. Individual sweeps across the jet axis (points) as well as their ensemble averages (solid black lines) show a coherent response, with most of the attenuation taking place in a narrow region of sensor's angular deflections. Because this attenuation can be related to the presence of a quantized vortices' tangle, we use the obtained data to calculate the corresponding vortex line density and probe its spatial structure. Additionally, as shown in the bottom panel of Fig. 4, the fluctuations of the second sound amplitude introduce a statistical uncertainty of around 10 %, which is comparable to the uncertainty one can associate to Eq. ( 6), as mentioned above -a similar level of fluctuations is observed for other axial distances and data sets A1-A2. Finally, no systematic error is apparently introduced by the stepper motor that displaces the sensor back and forth across the signal minimum, thanks to its positioning accuracy.

An approach similar to that used above is not possible for data sets in categories B and C because, for the corresponding temperatures, c 2 is strongly temperature dependent. Therefore, for each sensor position, a full frequency sweep across the resonance frequency (approx. 38.5 kHz) is measured and we estimate the corresponding resonance amplitude from the in-phase (A x ) and quadrature (A y ) components of the acquired signal, VJSS19,WVR23 since the plot of A y as a function of A x can be fitted with a circle having diameter equal to A. As in the previous case, the values of A 0 and ∆ 0 are taken from full fits of the resonance curves measured far away from the jet. Following this approach, we obtain a single value of A for each position of the second sound sensor and the statistical fluctuations of the sensor response are therefore not available for these data sets.

While sets B1-B4 probe the jet in both radial and axial directions and can then be directly compared to A1-A3, the set C1 only captures the vortex line density along the jet axis, i.e. for a constant angular deflection equal to α 0 .

Finally, the data set D1 refers to the flow visualization study, whose bath temperature and heating power can be directly compared with sets A1 and A2. The data set comprises approx. 1.88 million particle position-velocity pairs, within the above mentioned two-dimensional field of view.

For the measured values of T and P , see Table I, one can use Eqs.

(1-3) to calculate nominal values of v n , v ns and Re, assuming that the heat produced in the heater enclosure is transported towards the helium bath solely through the nozzle. In this idealized case, we obtain normal fluid and counterflow velocities of the order of 1 m/s, corresponding to Re values of the order of 10 5 . These values are reported in Table I.

However, our 3D-printed enclosure is subjected to heat leaks, through its plastic walls and the joint between the enclosure and brass nozzle. The leaks are found to substantially reduce the flow velocity and the Reynolds number of the jets studied below. This claim can be directly verified via the visualization data set, shown in Fig. 5.

In the left panel, we plot the mean particle velocity in the axial direction (color lines) as a function of the particle horizontal position in the field of view, corresponding to the radial direction. The dependence on the axial coordinate is specifically expressed by five offset lines, i.e. the 4d high field of view is split into 5 horizontal stripes, 0.8d high. At first glance, we can already say that the particle motions capture a spatially constrained flow structure. Outside the structure, the particles are practically still, while, inside it, they reach peak velocities of the order of 0.1 m/s. This value is in sharp contrast with the nominal values of v n reported in Table I, which are about one order of magnitude larger than the experimental observation. Therefore, the actual normal fluid and counterflow velocities are at least one order of magnitude smaller than the maximum values estimated with Eqs. (1-3).

Heat leaks can be taken into account by assuming a finite efficiency η < ∼ 0.1 of the heat transfer through the nozzle. In particular, the heat flux through the nozzle q in Eq. (1) should become ηq, which means that the effective heating power is reduced to ηP in comparison to the values reported in Table I. However, providing a clear estimate of η based on the current data is not straightforward, also considering that the visualized region is not directly located above the nozzle, but approximately 10 mm away from it -note that in a previous study vLM22 it was found that, for an analogous set-up, η ≈ 1/30. We can nevertheless use the upper limits of v n , v ns and Re reported in Table I to compare individual data sets, if we assume that η does not significantly change over the course of the experiment.

In order to investigate heat leaks in more detail, we placed a small germanium thermometer inside the enclosure and measured the temperature difference that develops between the enclosure and the bath after the heater is switched on. Time traces of the recorded temperature are presented in the left panel of Fig. 6 for a constant bath temperature of 1.30 K (note that the temperature difference is more prominent at lower temperatures because the heat capacity of He II is roughly proportional to p n in the investigated temperature range). Specifically, for a set of rising heating powers, we observe that the heater enclosure warms up and reaches a steady state within a few seconds after the heater is powered. Once the heater is switched off, the enclosure quickly thermalizes with the surrounding bath.

First of all, it is apparent that the steady temperature difference ∆T between the enclosure and the 4 He bath is significant, of the order of 0.1 K. Additionally, the right panel of Fig. 6 displays ∆T as a function of the heating power P . We observe a neat linear dependence between these quantities, similarly to Ref. MYN89, where it was also reported that, for counterflow jets, ∆T ∝ q 3 for q > ∼ 5 kW/m 2 . The latter is consistent with the FIG. 5. Left panel: mean axial velocity of our solid particles, visualized in five regions, 0.8d high, crossing the jet axis (the curves are offset by 40 mm/s relative to each other). The distance separating each region center from the nozzle is specified next to each curve. A positive velocity indicates that the particles move, on average, away from the nozzle. Color lines: experimental data; pale color areas: one standard deviation intervals; black lines: Gaussian fits, Eq. ( 9); see also Sec. III A. Right panels: centerline velocity v0 (top panel) and jet width wv (bottom panel) as a function of the distance z from the nozzle, obtained from the Gaussian fits. Note that the quantities x, wv and z are normalized by the nozzle diameter d. FIG. 6. Left panel: temperature inside the heater enclosure as a function of time. The bath temperature is kept constant at 1.30 K and the heater is supplied by a constant heating power (specified near each curve) from approximately zero time. Right panel: steady temperature difference ∆T between the enclosure and the 4 He bath as a function of the heating power P .

data reported in Table I if we take into account the above mentioned heat leaks, i.e. the results plotted in the right panel of Fig. 6 confirm that η < ∼ 0.1 in the range of investigated parameters because, otherwise, we should also see here the cubic scaling typical of turbulent counterflow in channels. GM49 Nevertheless, the observed temperature difference leads to the formation of a prominent jet that features a dense tangle of quantized vortices. We now investigate this tangle using the second sound sensor as described above. FIG. 7. Radial profiles of the vortex line density for P ≈ 776 mW and different temperatures. The colors correspond to different axial positions of the sensor, from 9d above the nozzle (dark blue) to 31.5d, 29d and 34d (dark red) for the left, middle and right panels, respectively, with steps of 2.5d. The curves are offset by 3 × 10 9 m -2 for the sake of clarity. Positive and negative values of the radial coordinate r, here normalized the nozzle diameter d, denote the corresponding angular deflection (r/d < 0 means < α0 and vice versa).

A. Spatial structure of the quantized vortices' tangle

The observed decrease of the second sound amplitude, indicating the presence of quantized vortices in the flow, depends on the radial and axial positions of the sensor, revealing the spatial structure of the quantized vortices' tangle associated to the counterflow jet. Fig. 7 displays the vortex line density L, also named VLD below, calculated from Eq. ( 6), for data sets A3, B3 and B4. These data sets are characterized by the same heating power (P ≈ 776 mW) and different temperatures. The individual curves correspond to different axial distances, starting from 9d (dark blue) above the nozzle, up to 31.5d, 29d and 34d (dark red), for sets A3, B3 and B4, respectively -the curves are vertically offset for clarity. These radial VLD profiles map the quantized vortices' tangle in unprecedented detail, and we specifically observe the tangle to be spatially confined and growing in the direction -considering the errors discussed above, we remind that individual L values are here given with relative uncertainty equal to approximately 10 %.

Moreover, the highest L values are obtained for the smallest axial distances and for the lowest temperature. A similar temperature dependence is displayed by the normal fluid velocity v n for a constant heating power, i.e. v n increases as T decreases -see Eq. ( 1). On the other hand, v ns is not a monotonous function of temperature and, for the displayed data, the smallest v ns is obtained for 1.95 K -see Eq. ( 2) and Table I. Specifically, this discrepancy between the temperature dependencies of VLD and v ns tells us that the physical mechanisms resulting in the formation of counterflow jets are different from those taking place in ordinary channel counterflow, where the scaling L ∝ v 2 ns was consistently observed for a rather wide range of temperatures and counterflow velocities. VBS15 In other words, our data support the picture that the quantized vortices' tangle associated to the jet is not simply generated in the nozzle, where channel counterflow takes place, and subsequently ejected into the volume above it. In particular, as discussed below, we find the quantized vortices' tangle to possess properties that are usually attributed to turbulent coflow of He II, i.e. to a configuration where the two fluid components are locked together in a single velocity field.

The obtained VLD profiles can be fitted with a Gaussian peak of the form where the peak amplitude L 0 and width w are taken as fitting parameters to match the experimental data. The Gaussian-like shape of our VLD profiles is reminiscent of the bell-shaped profile of the vorticity magnitude inferred in a classical round jet using the particle imaging velocimetry technique, WL98 and it is equally reminiscent of the bell-shaped profile of the energy dissipation inferred from hot wire anemometry measurements in a similar flow. HCG94 Note in passing that, under the assumption of homogeneous and isotropic turbulence, energy dissipation and squared vorticity are proportional to each other, Pop00 but, at the same time, one should keep in mind that quantities having similar Gaussian-like statistical distributions are not necessarily related from a physical point of view.

L(r) = L 0 exp - r w 2 , (8)
It is now useful to mention that the profiles of the axial velocity, presented in the left panel of Fig. 5, display a spatial structure similar to that of the VLD profiles. In direct analogy with Eq. ( 8), we then fit individual velocity profiles using

v(x) = v 0 exp - x w v 2 , ( 9
)
where the fitting parameters are the centerline velocity v 0 and the jet width w v . Specifically, the left panel of Fig. 5 shows the close agreement between the experimental data (color lines) and the fitting function (black lines), and the axial dependence of both fitting parameters is displayed in the two panels on the right. Additionally, to highlight the effect of our sensor motion on the investigated flow, we compare in Fig. 8 the L 0 values obtained by fitting the full VLD profiles of data set B4 (red crosses) with the directly measured peak values of data set C1 (points) as a function of z -note that data set C1 includes two axial strokes of the sensor. The L 0 values detected by the sensor that gradually moves towards the nozzle are marked by cyan points and the magenta points denote data taken by the sensor that moves in the opposite direction. The curves closely overlap within the full range of investigated distances and within the accuracy of the experimental technique, which neatly illustrates that our observations are fully reproducible.

B. Scaling of the vortex line density

We now analyze how the peak amplitude L 0 and width w of individual VLD profiles scale with the axial distance z. First of all, the relation between L 0 and z is presented in Fig. 9 for data sets A1-A3 and B1-B4, marked by open symbols (we also use this notation in the following figures). Here, no artificial vertical shift is imposed and the vertical distribution of data lines is given by different experimental conditions, listed in Table I.

The observed L 0 (z) dependencies are compatible with power laws, with exponents ranging from -2 to -3/2, as indicated by the black lines in the figure. Specifically, the former exponent is found to match relatively small peak densities, while the latter one fits relatively dense tangles. Since the centerline velocity v 0 ∝ z -1 holds for both classical and counterflow jets, LC03,MTMN08 it apparently follows that L 0 ∝ v 3/2 0 for the largest line densities, while the L 0 ∝ v 2 0 dependence holds for the smallest ones observed in our experiment. Two alternative interpretations of this puzzling behavior are discussed in Sec. IV.

Note in passing that our visualization data (see the top right panel of Fig. 5) do not seem to reproduce the expected scaling of the centerline velocity, although the latter does show the tendency to decrease as the axial distance increases. The mismatch can be attributed to the fact that the jet flow was visualized in the near field region, extending up to approximately 7d away from the nozzle, BFP12 where the mentioned scaling law is actually not expected to hold.

The axial dependence of the peak width w is plotted in Fig. 10. The data points characterized by a relatively large VLD amplitude display a neat linear dependence of w with z, which is also independent of experimental parameters, see colored points in Fig. 10. We identify these well-resolved VLD peaks by setting a threshold L t = 2.2 × 10 9 m -2 . This value is chosen so that L min /L t ≈ 0.05, i.e. it holds that the profiles meeting this threshold density display VLD values larger than L min , at least within the interval [-2w, 2w], which is required for a well-converged Gaussian fit and a reliable estimation of the fitting parameters. Additionally, the errors associated to the w estimates, which are computed from 95 % confidence bounds of the fit, range from 2 to 6 % for the well-resolved data, and the similarly obtained errors of L 0 are between 1 and 5 %. For the data sets that fail to meet the threshold value L t , the relative errors of w estimates are as high as 33 % -and those of L 0 are as high as 22 %. In the following, for the sake of brevity, the colored points in Fig. 10, those with L 0 > L t , are named high (data) set, while the others are called low (data) set, because, as already noted, they can be respectively associated to high and low values of peak density -see again Fig. 9.

We fit the widths corresponding to the high set together using

w(z) = β L (z -z 0 ) , (10)
where we find that the virtual origin z 0 ≈ -0.9d and that the growth rate β L = dw/dz of the vortex tangle is equal to approximately 0.14 (see the solid black line overlaying the FIG. 10. Peak width w as a function of the axial distance z (both quantities are normalized by the nozzle diameter d). Color symbols: points with L0 > Lt, i.e. data sets A2 (in part), A3, B3, and B4, collectively named high set; gray symbols: remaining points, with L0 ≤ Lt, i.e. data sets A1, A2 (in part), B1 and B2, collectively named low set. Note that the symbols are as in Fig. 9, regardless of their color; black line: linear fit, Eq. ( 10).

experimental data in Fig. 10). Similar values are reported for the velocity field of classical round jets. For example, the jet virtual origin is found to be rather close to the nozzle also in Ref. HCG94. Additionally, Ref. WF69 reports a linear growth of the velocity profile, with a rate β = 0.11, for Re ≈ 10 5 , and in Ref. ATK17 a similar value is given for Re ≈ 10 4 . For the sake of comparison, it is also useful to estimate the full width at half maximum (FWHM) of the VLD profile, and, for a Gaussian peak, we obtain β FWHM L = 2 √ log 2β L ≈ 0.23, which, once more, is a value comparable to the FWHM-based growth rate of approx. 0.21, reported in Ref. ZTBV22. However, one should keep in mind that velocity and vortex line density profiles are expected to be different in nature, e.g. the widths of velocity and VLD profiles are not expected in general to grow at the same rate, and, on top of this, counterflow and classical jets are characterized by different production mechanism, which should shift the virtual origin. Note, finally, that the bottom right panel of Fig. 5 presents the width of the velocity profiles versus the axial distance z, obtained from our flow visualization measurements in He II. Here again we find an affine growth of the velocity profile width, consistent with that of classical jets.

In summary, the emergence of approximately Gaussian profiles for the vortex line density, alongside with the scaling laws outlined above, suggests that the quantized vortices' tangle develops a structure that is tightly coupled to the associated velocity field, at least within the intermediate field region of the jet. In other words, the quantized vortices' tangle grows in size, while decreasing its peak density with z, similarly as the peak flow velocity that decreases with z, due to turbulent entrainment and momentum conservation.

IV. DISCUSSION

In the following, we outline two interpretations of the just presented experimental results. The first one, detailed in Sec. IV A and proposed by P. Švančara and P.-E. Roche, leads us to conjecture a self-similar functional form of the vortex tangle density across counterflow jets. The other, detailed in Sec. IV B and proposed by M. La Mantia, suggests that the position of the superfluid stagnation point could depend on the flow forcing as well. Both interpretations can be used to explain in part the obtained results.

A. Self-similar jet model

It is apparent from Figs. 9 and 10 that the curves associated to the smallest peak densities, corresponding to the low set, defined in Sec. III B, are characterized by a larger scatter in comparison to the other curves. This can be attributed to the less accurate fit of Eq. ( 8) to weakly attenuated second sound signals, i.e. to relatively low values of L, especially in the off-axis region. In view of improving the statistical convergence, we introduce here an alternative, dimensionless quantity, L z , which aggregates a larger set of measurements. Specifically, if we assume that the distribution of quantized vortices in the jet is axially symmetric, i.e. that L is independent of the azimuthal coordinate φ, we calculate this quantity, the cumulative length of quantized vortices per unit height, as

L z (z) = ∞ 0 2π 0 L(r, z)rdrdφ = ∞ 0 2πrL(r, z)dr. (11)
Additionally, we note from Fig. 10 that the VLD profiles associated to the high set are self-similar, at least in the range of investigated parameters. This self-preserving property of jets far away from the nozzle is well-known for classical jets Pop00 and it was also invoked to interpret previous experiments on counterflow jets. LL84 This implies that the (high set) VLD profiles can be modeled as

L(r, z) = G r z -z 0 L 0 (z) = G(x)L 0 (z), (12)
where the dimensionless profile shape G(x) can be approximated, for example, by a Gaussian peak, ensuring that G(0) = 1. Eqs. ( 11) and ( 12) then lead to the scaling relation

L z (z) = (z -z 0 ) 2 L 0 (z) ∞ 0 2πxG(x)dx ∝ z 2 L 0 (z), (13)
where we assume that the integral is finite and that z 0 z. Eq. ( 13) shows us how L z and L 0 are related under the self-similarity hypothesis.

The cumulative vortex length L z can be estimated from the experimental data in two different ways. The first one consists in using the full VLD profiles, e.g. those presented in Fig. 7. We can then numerically estimate the right-hand side of Eq. ( 11) within the limits (r min , 0) or (0, r max ), where r min (r max ) is the minimum (maximum) radial coordinate probed by the sensor. Since the obtained radial profiles fully capture the region occupied by the quantized vortices' tangle, as shown, e.g. in Fig. 7, these integrals adequately approximate Eq. ( 11) -note that, to reduce error, we take as the final estimate of L z for this first method the average value of the integrals with r min and r max . The second approach relies on the explicit form of G introduced above to model the high set profiles. In agreement with Eqs. ( 8) and ( 10), we set 14) and evaluate the integral on the right-hand side of Eq. (11) as

G r z -z 0 = exp - r β L (z -z 0 ) 2 = exp - r w 2 (
L z = 2πL 0 ∞ 0 exp - r w 2 rdr = πL 0 w 2 . ( 15
)
Both estimates of L z are presented in Fig. 11 as a function of the axial distance z. The left panel represents the right-hand side of Eq. ( 15) and the right panel displays the result 15), computed using the experimentally obtained values of L0 and w. Right panel: result of the numerical integration of the right-hand side of Eq. ( 11) as discussed in the text. Colors as in Fig. 9.

of the numerical integration of the right-hand side of Eq. ( 11) using the trapezoidal rule. The two methods yield comparable results except for data set A1 (open blue circles, right panel), where the nonphysical, negative values of L z most likely result from the poor signalto-noise ratio (a negative vortex line density is obtained from Eq. ( 6) when A > A 0 ). Note again that Eq. ( 15) is derived assuming that the studied flows are self-similar.

It is apparent from Fig. 11 that, for a given heating power P and helium bath temperature T , i.e. for a given data set, the cumulative length L z does not depend on z within the experimental uncertainty and for all data sets but one. Taking Eq. ( 13) into account, we can therefore say that L 0 (z) ∝ z -2 in the range of investigated parameters, excluding data set A3 (yellow triangles). In this case, L z grows with the axial distance z, which is an unexplained exception. Nevertheless, leaving this data set aside for the moment, it now seems that the variations of the power-law exponents apparent in Fig. 9 are compensated by suitable changes in the growth of the measured VLD profile width, presented in Fig. 10. This compensation could be explained, at least in part, by stochastic fluctuations of the jet direction. The latter could systematically decrease the measured centerline density of quantized vortices, and simultaneously increase the measured jet width. Apparently, the use of the cumulative vortex length allows us to largely discard the data scatter observed in Figs. 9 and 10.

The inverse quadratic scaling of L 0 (z) has a straightforward interpretation using a heuristic relation tested by numerical simulations and experiments in steady, nearly-homogeneous coflow of turbulent He II. SRL11,BVS + 14 These works relate L c , the mean density of the quantized vortices' tangle in coflow, and the energy dissipation rate ε as

L 2 c ∝ ε ∝ v 3 rms H , (16)
where v rms is the root mean square velocity of the coflow being investigated and H indicates a velocity correlation distance usually called the integral length scale.

Turbulence along the axis of classical round jets can be locally approximated as homogeneous and isotropic, Pop00 and some local properties of turbulence can be inferred from the local values of v rms and H, as shown, for example, in Refs. ASZ96, CCCH00, and IKY + 03. Specifically, following this reasoning one can estimate along the jet axis the vorticity magnitude and the energy dissipation rate, appearing in the classical relation analogous to Eq. ( 16). In other words, we assume here that Eq. ( 16) holds locally on the counterflow jet axis, with L c ≡ L 0 (note that in Ref. LL84 the assumptions of local homogeneity and FIG. 12. Mean cumulative length Lz z as a function of the heating power P . Colors as in Fig. 9; error bars correspond to one standard deviation interval. Black dotted lines connect data points collected at the same temperature of the helium bath.

isotropy have been already used to interpret measurements of second sound waves crossing a jet).

In particular, for a given data set, and taking into account the scaling relations v rms v 0 and H ∝ w ∝ z, Eq. ( 16) becomes

L 2 0 ∝ v 3 0 w ∝ z -3 z ∝ z -4 , (17)
which fully accounts for the observed inverse quadratic scaling of L 0 with z. Additionally, for a fixed temperature, we can write that

L 0 (z, P ) ≈ z -2 F(P ), ( 18
)
where F is an unknown function of the heating power. Now, in view of further reducing our data scatter, we take advantage of the conjectured invariance of L z along the jet axis and we calculate its average value -the averaging operator is denoted as • z and the average value is therefore L z z . Fig. 12 shows L z z , computed from the data displayed in the left panel of Fig. 11, as a function of the heating power -the dotted lines connect points corresponding to the same bath temperature. Although the data scatter -likely due to experimental errors and temperature dependence -is not negligible, we find that L z z roughly follows a P 3/2 scaling (black line) that ought to be interpreted. Assuming that the dimensionless VLD profile shape G is independent of P , we find from Eqs. ( 13) and ( 18) that

L z z ∝ z 2 L 0 (z, P ) z ≈ F(P ). ( 19
)
What now remains is to understand why F ∝ P 3/2 . At any given z, it results from the proportionality between L 2 0 and v 3 0 in turbulent coflows, SRL11,BVS + 14 which is apparent from Eq. ( 17). Due to the counterflow generation mechanism, Eq. (1), the centerline velocity v 0 scales linearly with P (or ηP to account for heat losses). We therefore obtain F ∝ L 0 ∝ v 3/2 0 ∝ P 3/2 , consistently with the outcome of Fig. 12. It is now useful to mention Ref. Lag75, where an elongated counterflow jet of He II was investigated at a distance of approx. 10 nozzle widths from the flow source using a second sound beam traveling across the jet. For temperature (1.6 to 2.1 K) and heat flux (3 to 50 kW/m 2 ) ranges comparable to those of the present work, the attenuation factor, which is proportional to the vortex line density, was reported to scale as P k with k = 1.3-1.6. The outcome was later explained using geometrical acoustics, assuming that the counterflow jet is turbulent and that the normal and superfluid components are locked in a single velocity field. LL84 To summarize, excluding data set A3 that remains to be understood, our results are consistent with a self-similar spatial distribution of the quantized vortex tangle, whose density L can be written as

L(r, z, P ) ∝ P 3/2 z 2 G r z -z 0 , ( 20
)
where G is a Gaussian-like function. If we substitute for G the employed Gaussian fit, we obtain an explicit functional form, i.e.

L(r, z, P

) ∝ P 3/2 z 2 exp - r β L (z -z 0 ) 2 , ( 21
)
where the z -2 factor accounts for the linear development of the integral scale along the jet, far away from the nozzle. Now, for consistency with the exponential term, the jet virtual origin can be reintroduced in this factor, to better account for smaller z, see Eq. ( 13).

Additionally, although the determination of the temperature dependence is not possible on the basis of the present data, three physical arguments suggest how to make it explicit. First of all, the spreading rate of classical jets is due to the balance between mixing and axial entrainment, and these processes should remain equally efficient in He II when both components are flowing together. Therefore, we expect that β L will have a weak dependence on T and P . On the contrary, the virtual origin z 0 can be associated the crossover from the to the coflow regime, and we then expect it to explicitly depend on T and P . Finally, the amount of quantized vortex lines generated in a given He II coflow depends on temperature, because the vortex dynamics depends on the normal fluid fraction, Lag75 which is strongly temperature dependent, see again Fig. 1 -one can then embed the latter consideration into an unknown function of temperature, T .

Following the previous reasoning, we can consequently write a generalized version of Eq. ( 21), i.e. L(r, z, P, T ) ≈

P 3/2 [z -z 0 (P, T )] 2 exp - r β L [z -z 0 (P, T )] 2 T (T ), ( 22
)
where z is large compared to z 0 , to meet the self-similarity condition, β L ≈ 0.14, and z 0 ≈ -0.9d, as reported above. We remind, however, that the numerical values might be biased by the response of the second sound sensor to the flow velocity; that is, we assume in this work that the contribution of the fluid velocity to second sound attenuation is not significant. More generally, further studies are required to quantitatively validate the proposed model, e.g. to understand why the L z versus z scaling differs for data set A3 compared to the others.

B. Moving stagnation point interpretation

In Sec. III B we reported that the L 0 (z) dependencies plotted in Fig. 9 are compatible with power laws, with exponents ranging from -2 to -3/2. Specifically, the former exponent can be associated to relatively small peak densities, i.e. to the low set, while the latter one is more suitable for relatively denser tangles, i.e. for the high set. Considering that v 0 ∝ z -1 is expected to hold for self-similar round jets, one can then derive L 0 ∝ v 3/2 0 for the largest line densities, while the L 0 ∝ v 2 0 scaling is more adequate for the smallest ones. As already mentioned, the 3/2-power scaling was reported in previous studies -see e.g. Refs. LL84 and VBS15 -and can be associated to turbulent coflow of He II. More precisely, one can derive it from Eq. ( 16) assuming that the integral length scale H does not depend locally on other flow scales. This could happen in the region where the hypothesis of locally homogeneous and isotropic flow is expected to hold, i.e. along the jet axis, where no information on the jet width w should be accessible, if the typical size of this region is smaller than w -note again that H ∝ w for self-similar jets. In other words, the assumption made in this section is that the integral length scale H is constant in the region where Eq. ( 16) holds.

The 2-power scaling is instead often associated to channel counterflow, i.e. L ∝ v 2 ns , discussed e.g. in Ref. However, as already noted in Sec. III A, the normal fluid and counterflow velocities are in general not directly proportional to each other, but this is only true at liquid temperatures higher than approximately 2 K. DB98 It then follows that, at lower temperatures, v n ∝ v ns . Additionally, considering that, as reported in Sec. IV A, v 0 should be proportional to v n , we find that L 0 ∝ v 2 0 for our more dilute tangles, i.e. for the low set, collected at temperatures lower than 2 K.

From this discussion one can then say that, at sufficiently low values of heating power, channel counterflow features could also be observed in counterflow jets. The physical reason could be related to the mutual friction force, which can be seen as an energy sink. Specifically, this dissipation mechanism could become inefficient for counterflow jets at sufficiently high values of heating power. In other words, the position of the stagnation point depicted in Fig. 2 could be a function of the heating power and move closer to the nozzle as the power increases.

The suggestion appears a bit less adventurous if one considers that counterflow jets at relatively low heating power have yet to the thoroughly investigated, or at least they have received to date less attention than those at higher heating power. Indeed, we reported in Sec. III that, on the basis of Ref. MYN89, we expect ∆T to be linearly proportional to q in the range of parameters probed here, while this is not the case of other studies on counterflow jets, such as Refs. MI89 and DB73, where a neat cubic scaling was observed at larger values of heat flux. Specifically, Refs. MYN89 and NM94 are apparently the only works where some attention is given to the just mentioned linear regime and, more importantly, the claim that the superfluid stagnation point is located in the close proximity of the nozzle is solely based on data collected in the cubic regime, i.e. at relatively high values of heating power.

Finally, it is useful to mention that, from the right-hand side of Eq. ( 15) and the L 0 ∝ v 3/2 0 scaling, one can get that L z should be proportional to z 1/2 , at least for the high set. However, in Fig. 11 we observe that L z grows with the axial distance z only for data set A3, corresponding to the largest Reynolds number, while this is not the case for the other data sets displaying the proportionality between w and z. Once more, further experiments are required to assess the validity of the physical interpretations presented here.

V. CONCLUSIONS

The intermediate field region of round counterflow jets, representing a unique type of thermally driven flow of He II, was studied in the turbulent regime, for Reynolds numbers of the order of 10 4 , using the second sound attenuation and flow visualization techniques. Specifically, a miniature second sound sensor was employed to detect the vortex line density across the jet axis, with a spatial resolution slightly larger than the nozzle diameter, at fluid temperatures between 1.64 and 2.10 K -note, however, that the measured density is biased by an additional sensitivity of the employed second sound sensor to the flow velocity, which cannot be precisely quantified at the moment.

Nevertheless, within the range of investigated axial distances, from 9 to 34 nozzle diameters, we observe a highly inhomogeneous tangle of quantized vortices. In particular, the vortex line density displays a Gaussian-like profile in the radial direction and, for the highest density values, its width w grows linearly with the axial distance z at a rate dw/dz ≈ 0.14, which is comparable but not equal to the radial growth rate observed for the classical velocity field -similar remarks apply to the jet virtual origin. Additionally, as shown in Fig. 9, the peak density of the quantized vortices' tangle appears to follow, as a function of the axial distance, power laws with exponents ranging approximately between -2 and -3/2. Two physical interpretations of the observed scaling behavior are proposed, and none of them account for all the measured data sets. One interpretation conjectures a general functional form of the vortex line density in counterflow jets, which depends on spatial coordinates and heating power. The other suggests that the position of the superfluid stagnation point along the jet axis could depend on the heating power as well.

In summary, the present work indicates that future studies on counterflow jets should be devoted to extend the ranges of experimental parameters accessed to date, e.g. in order to quantitatively identify the conditions in which jet flows of superfluid 4 He are appreciably different form their classical analogs.

FIG. 3 .

 3 FIG. 3. Sketch of the experimental setup (not to scale). Brown: 3D-printed heater enclosure; blue arrows: possible motions of the mobile shaft; red: sensitive area of the second sound sensor. Dimensions in the right panel are in millimeters.

FIG. 4 .

 4 FIG. 4. Top panel: two sets of 30 sweeps (points) across the jet axis, obtained at 1.65 K and 776 mW (data set A3) for z/d = 9.0 and 31.5, as indicated in the figure. Solid and dashed black lines: ensemble-averaged data used for the calculation of the quantized vortices' density. Dot-dashed black line: deflection offset α0 ≈ 1.9 • , Eq. (7). Bottom panel: residuals (points) and standard deviations (lines) corresponding to the data displayed in the top panel.

FIG. 8 .

 8 FIG.8. Peak vortex line density (i.e. density along the jet axis) L0 as a function of the axial distance z, normalized by the nozzle diameter d, for data sets B4 (red crosses) and C1 (cyan and magenta points). Color arrows denote the direction of the sensor stroke (receding from or approaching towards the nozzle).

FIG. 9 .

 9 FIG. 9. Peak vortex line density L0 as a function of the normalized axial distance z/d. Data set labels are indicated to the left of each curve. Note the log-log scale.

FIG. 11 .

 11 FIG. 11. Cumulative length Lz of quantized vortices per unit height as a function of the normalized axial distance z/d. Left panel: right-hand side of Eq. (15), computed using the experimentally obtained values of L0 and w. Right panel: result of the numerical integration of the right-hand side of Eq. (11) as discussed in the text. Colors as in Fig.9.

  FIG. 2. Sketch of the two-component flow field in a thermal counterflow jet. Red (blue) arrows: normal (superfluid) component; green point: stagnation point of the superfluid component. Displayed are also velocity profiles in the near field, i.e. in the close proximity of the nozzle (bottom) and in the intermediate field (top), see Eq. (4). Adapted from Ref. NMK94.

TABLE I .

 I Summary of experimental conditions. T : temperature; P : heating power; vn/η: normal fluid velocity inside the nozzle, Eq. (1); vns/η: counterflow velocity inside the nozzle, Eq. (2); Re/η: Reynolds number, Eq. (3). Note that the calculated values for the velocity and the Reynolds number can be seen as maximal theoretical values, obtained by assuming no heat leaks from the heater enclosure. In reality, the efficiency of the heat transfer through the nozzle is η <

	in the text.				∼ 0.1, as specified
	Label T [K] P [mW] vn/η [m/s] vns/η [m/s] Re/η [10 3 ]
	A1 1.65	194	0.78	0.97	175
	A2 1.65	381	1.51	1.97	338
	A3 1.64	776	3.24	3.98	726
	B1 1.95	194	0.26	0.50	55
	B2 1.95	381	0.52	0.99	108
	B3 1.95	776	1.03	2.00	215
	B4 2.10	776	0.65	2.40	107
	C1 2.10	775	0.65	2.40	106
	D1 1.66	240	0.92	1.15	206

ACKNOWLEDGMENTS

We thank Jerôme Valentin for microfabricating parts of the employed second sound sensor. We are also thankful to Eric Woillez and the technical staff of Institut Néel for their prompt assistance during the second sound experiments, performed in Grenoble by P. Švančara and P.-E. Roche -the visualization and thermometry experiments were performed in Prague by P. Švančara and M. La Mantia. P.-E. Roche acknowledges support from the ANR QUTE-HPC grant (ANR-18-CE46-0013-03) and the EU Horizon 2020 program The European Microkelvin Platfom (EMP 824109). P. Švančara and M. La Mantia were supported in part by the Czech Science Foundation (GA ČR) under grant 19-00939S. P. Švančara also acknowledges the Barrande Fellowship Programme supporting doctoral mobility. Additionally, during the manuscript preparation in 2022, M. La Mantia was in part supported by the Czech Science Foundation (GA ČR) under grant 20-00918S.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

All authors contributed equally to this work.


Lénaïc Madec
email: lenaic.madec@univ-pau.fr

Jean-Bernard Ledeuil

Julien Morey

Hervé Martinez

Cross-section nano-Auger/SEM analysis to reveal bulk chemical/morphological properties of composites for energy storage

Composite materials for energy storage such as powders, electrodes or battery stack often require probing their bulk chemical/morphological properties, which remains challenging so far with conventional analytical methods. In this work, Ar + milling cross-section is proposed to reveal the intrinsically buried bulk information of three different composites without physical/chemical change. Then, nano-Auger/scanning electron microscopy (SEM) analysis is proposed to investigate their bulk properties at both micro-and nano-scales. For MnCo-based powders with micrometric particles, it allowed revealing the bulk porosity and the bulk nano-or micro-Mn/Co distribution. For micrometer thick TiSnSb-based electrodes, it allowed proving the conversion reaction over long term cycling (i.e. the participation of the electrochemically inactive Ti) while revealing the TiSnSb particles morphological evolution (shell to core spreading/pulverization into porous structure) and SEI formation inside the porous TiSnSb. For PEO-based solid battery stacks, the cross-section allowed revealing well-defined interfaces so that reliable interfaces analysis can thus be perform. Advantage/limitation of this cross-section nano-Auger/SEM approach are also discussed. Overall, this work opens the door for future development of Ar + milling cross-section and Auger analysis as powerful tools to reveal/study buried chemical/morphological properties at micro-and nano-scales even beyond the energy storage field.

Introduction

Composite materials and electrodes for energy storage often require probing their bulk chemical/morphological properties. However, conventional analytical methods are often limited to do so as the targeted information is intrinsically buried in the bulk of composites.

For instance, pseudocapacitive composites that have attracted interest to develop high energy supercapacitors, [START_REF] Zhang | Graphene-based materials as supercapacitor electrodes[END_REF] , [START_REF] V. Augustyn | Pseudocapacitive oxide materials for high-rate electrochemical energy storage[END_REF] especially for binary MnO2-based composites that try to combine the high capacity of MnO2 [START_REF] Toupin | Charge storage mechanism of MnO2 electrode used in aqueous electrochemical capacitor[END_REF] to the high electronic conductivity of the second components (carbons, metals, metal oxides/hydroxides or polymers). [START_REF] Wei | Manganese oxide-based materials as electrochemical supercapacitor electrodes[END_REF] , [START_REF] Wang | Engineering of MnO<inf>2</inf>-based nanocomposites for high-performance supercapacitors[END_REF] However, some questions still remain difficult to answer, especially in the case of nano-composites with a micrometric particles size: (i) What is the bulk morphological organization, i.e. the distribution scale of the constituents within the composites? (ii) What is the bulk structure, i.e. the particles size and porosity within the composites? To answer these questions, X-ray diffraction (XRD) is often used to access the size of the coherent domain but it remain limited as it cannot give access to the phase organisation within the composites. Scanning electron microscopy (SEM) and transmission electron microscopy (TEM) coupled to energy dispersive X-ray spectroscopy (EDS) remains also limited as SEM-EDS probes the micro-scale while TEM-EDS probes individual and dispersed nanostructures unlikely representative of a micrometric sample.

Another example is conversion-based electrodes that have attracted interest to develop high-energy Li-ion batteries, [START_REF] Aravindan | Research Progress on Negative Electrodes for Practical Li-Ion Batteries: Beyond Carbonaceous Anodes[END_REF] especially for binary or ternary conversion materials that use an electrochemically inactive element to buffer the volume expansion issue (>200%). [START_REF] Aravindan | Research Progress on Negative Electrodes for Practical Li-Ion Batteries: Beyond Carbonaceous Anodes[END_REF] , [START_REF] Mukhopadhyay | Deformation and stress in electrode materials for Li-ion batteries[END_REF] However, some questions still remain difficult to answer, especially in the case of long term cycling of practical (μm thick) electrode: (i) what is the morphological evolution of the conversion material particles, i.e. how does the particles pulverization occur? (ii) Does the conversion reaction still occur? In addition, does the inactive element (if binary/ternary materials are used) still participate? (iii) Does the continuous solid electrolyte interphase (SEI) formation/breaking take place where the particles break up? In addition, can it be observed? To answer these questions, XRD [START_REF] Sougrati | TiSnSb a new efficient negative electrode for Li-ion batteries: mechanism investigations by operando-XRD and Mössbauer techniques[END_REF] , [START_REF] Ikeda | Behaviour of various cathode materials for non-aqueous lithium cells[END_REF] and X-ray absorption spectroscopy (XAS) [START_REF] Fehse | In-Depth Analysis of the Conversion Mechanism of TiSnSb vs Li by Operando Triple-Edge X-ray Absorption Spectroscopy: A Chemometric Approach[END_REF] can be used to study/confirm the conversion reaction mechanism but they remain often limited to the first lithiation/delithiation cycle due to the fast amorphisation of binary/ternary conversion materials. Ex situ [START_REF] Grugeon | Particle Size Effects on the Electrochemical Performance of Copper Oxides toward Lithium[END_REF] , [START_REF] Tarascon | New concepts for the search of better electrode materials for rechargeable lithium batteries[END_REF] , [START_REF] Yonekura | Progress of the conversion reaction of Mn3O4particles as a function of the depth of discharge[END_REF] , [START_REF] Wang | Conversion reaction mechanisms in lithium ion batteries: Study of the binary metal fluoride electrodes[END_REF] and in situ [START_REF] Liu | A pomegranateinspired nanoscale design for large-volume-change lithium battery anodes[END_REF] , [START_REF] Su | In Situ Transmission Electron Microscopy Observation of Electrochemical Behavior of CoS 2 in Lithium-Ion Battery[END_REF] TEM techniques have also been widely used to study the mechanism, volume change and SEI formation of conversion materials at the nanoscale but these techniques focus on individual nano-particles, which is not representative of the reactivity occurring in three-dimensional electrodes. In operando X-ray tomographic microscopy (SRXTM) was also proposed to simultaneously visualize in three-dimension chemical and morphological evolution of individual particles at the electrode scale [START_REF] Ebner | Visualization and Quantification of Electrochemical and Mechanical Degradation in Li Ion Batteries[END_REF] but it remains limited in term of chemical information (impossibility to analyze SEI components for instance) and it often requires a synchrotron radiation.

Finally, solid-state batteries appear very attractive to improve further the energy density with longer lifetime and improved safety at an affordable cost. [START_REF] Manthiram | Lithium battery chemistries enabled by solid-state electrolytes[END_REF][START_REF] Zhang | Recent advances in solid polymer electrolytes for lithium batteries[END_REF] However, the main current challenge to these batteries performance still remains to be overcome: How to reveal and analyze the numerous electrode materials/solid electrolyte interfaces and their electro-chemo-mechanical stability with high reliability as they are buried in battery stacks? In that direction, state-of-the-art cross-section preparation techniques such as focused ion beam (FIB) or broad ion beam (BIB, i.e. Ar + ion milling using a cross-polisher) have attracted interest to first reveal buried interfaces prior to their analysis by various analytical techniques. [START_REF] Zuo | A mechanistic investigation of the Li10GeP2S12|LiNi1-x-yCoxMnyO2 interface stability in all-solid-state lithium batteries[END_REF][START_REF] López | A critical discussion on the analysis of buried interfaces in Li solid-state batteries.: Ex situ and in situ / operando studies[END_REF] However, these cross-section approaches remain limited so far in the case of polymer-based solid-state batteries due to the high sensitivity/degradation of polymers under ion beams exposition without specific precaution.

To tackle these issues, this work is based on the following approach (Figure 1): (i) specific crosssection preparations using Ar + ion milling to access the bulk of composites via the formation of flat surfaces without chemical change coupled to (ii) nano-Auger/SEM analysis to investigate the composites bulk chemical/morphological properties at micro-and nano-scales. The advantage and limitation of this cross-section nano-Auger/SEM approach is discussed regarding different composites for energy storage, namely MnCo-based powders, TiSnSb-based electrodes and a poly(ethylene oxide)-based solid-state battery stack. Overall, this work highlights the interest of Ar + milling cross-sections preparation and opens the door for future development of Auger analysis as a powerful tool to study chemical/morphological properties at both micro-and nanoscales in the field of energy storage and beyond.

Figure 1. Schematic representation of the Ar + ion milling cross-sectioning and nano-Auger/SEM analysis approach followed to access and investigate the bulk chemical/morphological properties of composites at both micro-and nano-scales.

Materials and methods

Composites preparation

MnCo composite powders with a 1:1 Mn:Co ratio were prepared by exfoliation/restacking of starting H-MnO2 and β3-CoOOH [START_REF] Tang | Prediction of Isoelectric Point of Manganese and Cobalt Lamellar Oxides: Application to Controlled Synthesis of Mixed Oxides[END_REF] following previously described procedures. [START_REF] Tang | Synergy of Mn and Co in Slab-Based Nanocomposites for Hybrid Supercapacitors: Impact of Restacking Process on Electrochemical Properties[END_REF] TiSnSb synthesis, electrodes preparation (70:9:9:12 weight ratio of TiSnSb:carbon black:VGCF:CMC with 3 ±0.3 mgTiSnSb/cm -2 ) and cycling procedure are fully described elsewhere. [START_REF] Madec | Toward efficient Li-ion cells at high temperatures: Example of TiSnSb material[END_REF] For clarity, cycling was performed at 25°C using 1м LiPF6 EC:PC:3DMC + 5% FEC + 1% VC as electrolyte. Cells were stopped after the first lithiation at 0.02 V or after 400 cycles at 1.5 V. Before cross-section preparation, TiSnSb electrodes were washed twice by immersion in DMC (anhydrous, ≥99% purity, Aldrich, 1 ml) in a clean and dry glass vial for 10 s.

PEO-based solid-battery stack was prepared as followed: PEO (900000 g/mol, Aldrich) was first dried at 120°C under vacuum for 24h then PEO was dissolved in anhydrous acetonitrile (ACN, 99.8%, Aldrich) by magnetic stirring at 50°C for 48h under argon. Lithium bis(trifluoromethanesulfonyl)imide (LiTFSI, 99.95%, Aldrich) was then added with a 25 :1 OPEO:LiLiTFSI ratio and kept under stirring for 24h. The obtained gel was then infused into NMC532 electrodes (S4R Montpellier, ~11.5 mg/cm 2 , pressed at 2 tons, ~50 μm thick, ~30% porosity) and graphite electrodes (S4R Montpellier, ~6.1 mg/cm 2 , pressed at 2 tons, ~50 μm thick, ~30% porosity) deposited in CR2032 coin cell cases. The ACN was then slowly evaporated using a specific heating procedure (30 min at 30°C, 50°C, 60°C and 1h at 70°C), which allowed filling the entire electrodes porosity by the PEO-LiTFSI electrolyte. Note that an excess of gel was used so that the PEO-LiTFSI electrolyte not only filled the electrode porosity but also formed a thin film covering the electrodes. Finally, a full solid-state polymer-based battery stack was assembled by putting into contact the two PEO-LiTFSI based electrodes at 70°C for 24h.

Cross-section preparation of the composites using Ar + ion milling

Figure 2 top shows the schematic representation of the specific sample preparation followed prior to the cross-sectioning by Ar + ion milling as function of the composites. For all samples, preparation was performed in a nitrogen-filled glove-box. For MnCo, the powders were handmixed in a conducting silver-charged epoxy resin (to limit possible charging effect during Auger/SEM analysis) then deposited on a silicon wafer. Then the whole was fixed to a JEOL sample holder with the same epoxy resin. For TiSnSb, electrodes were sandwiched between two silicon wafers using a conducting silver-charged epoxy resin. Then the whole was fixed to a JEOL sample holder with the same epoxy resin. Finally, the MnCo and TiSnSb assemblies were then cross-sectioned at normal angle from the silicon wafer using a mask directly in contact with the silicon wafer. For PEO-based solid-battery, the stack was first thinned using a vertical slicer then deposited on a copper foil (GoodFellow, 99.97%, 100 µm thick) with a conductive carbon glue (Pelco). The whole was then fixed to a Technoorg LINDA sample holder with the same conductive carbon glue. Finally, the PEO-based assembly was then cross-sectioned at normal angle from the top of the assembly using a mask positioned at about 20 µm.

For MnCo composite powders and TiSnSb composite electrodes, cross-sections were obtained using a JEOL cross-Polisher (JEOL Ltd, Tokyo, Japan) working at 1.0 x10 -4 Pa with an Ar + beam of 6 keV, ∼120 µA (emission current) for 4h or 8h, respectively. For the PEO-based solid-battery stack, cross-section was obtained using a SEMPrep2 cross-Polisher (Technoorg LINDA) working at 1.4 x10 -4 Pa with an Ar + beam of 8 keV, 3.5 mA (i.e. 250 µA emission current) for 6h under liquid nitrogen cooling (133°K) to supress possible polymer degradation. Note that the use of the copper foil and the conductive carbon glue was also used to improve further the cooling of the assembly.

Overall, planar surfaces were obtained for all composites (Figure 2 bottom). However, Ar + milling is known to often lead to material re-deposition due to the sputtering process thus inducing possible curtaining effect, especially for samples with pores. Importantly, note that no chemical change is expected as Ar + milling is known to limit sample damage to amorphisation with the same chemical composition together with Ar + implantation [START_REF] Schuhrke | Investigation of surface amorphization of silicon wafers during ion-milling[END_REF] , [START_REF] Mccaffrey | Surface damage formation during ion-beam thinning of samples for transmission electron microscopy[END_REF] , [START_REF] Mitome | Ultrathin specimen preparation by a low-energy Ar-ion milling method[END_REF] that was never observed in our case (no Ar LMM Auger lines at 211 and 195 eV in KE).

Scanning electron microscopy (SEM), Auger electron spectroscopy (AES) and scanning Auger microscopy (SAM)

Importantly, the probe size for AES (i.e. to record an Auger spectrum) is about 20 nm leading to a spatial resolution for SAM (i.e. to record an Auger image) of about 25 nm while the probing depth is about 2-3 nm in the analysis conditions used. Thus, the simultaneous presence of different Auger transitions from different elements (Ti + Sn + Sb or Mn + Co for instance) in a spectrum means that these elements coexist, i.e. are located together in a 25 nm wide area with a 3 nm depth. SEM, AES and SAM were performed using a JEOL JAMP 9500 F Auger spectrometer (JEOL Ltd, Tokyo, Japan) equipped with a Schottky Field Emission gun and a hemi-spherical analyser coupled with a high dynamic multichannel detector. The operating pressure was <2 x10 -7 Pa and analysis were performed at 30-40° tilt to prevent charging effect except for thickness measurement that were done at 0° tilt. For Auger analysis, the electron beam was fixed at 10 keV/5 nA for MnCo and TiSnSb or at 15 keV/5 nA for the PEO-based stack. AES survey spectra were recorded with 1 eV step size using a focused probe and a CRR mode (constant relative resolution) with a constant relative energy resolution dE/E= 0.5% (high sensitivity). Auger spectra are presented using the EN(E) mode expressed as the output signal of the electron detector using "true" pulse counting versus kinetic energy. SAM (elemental 2D distribution images) were recorded using a CAE mode (constant analyzer energy) corresponding to a fixed energy resolution. SAM images are represented using "peak minus background" (P-B) Auger intensity for a transition (dE/E=0.5%). An "auto probe tracking" correction was also applied to control and compensate any potential drift during acquisition.

Results and discussion

MnCo-based powders -It is reminded that these composites (noted RMC1a, RMC1b and RMC2) were prepared from the exfoliation/restacking of H-MnO2 and β3-CoOOH following different procedures that are summarized on Figure 3 for better clarity. Note first that previous XRD analysis showed that the thickness of Mn-based domains decrease while it did not change for Cobased domains (except for RMC1a). [START_REF] Tang | Synergy of Mn and Co in Slab-Based Nanocomposites for Hybrid Supercapacitors: Impact of Restacking Process on Electrochemical Properties[END_REF] However, no conclusion was obtained from XRD about the phase organisation within the composites. Thus, SEM/nano-Auger analysis was performed on these composites cross-section to investigate the exact bulk structure (porosity) and bulk morphological organization (Mn / Co distribution scale), Figure 3. SEM images showed a highly porous bulk structure for RMC1a and RMC2 (Figure 3a and3c, respectively) in agreement with their high specific BET surface area (about 95 m 2 /g [START_REF] Tang | Synergy of Mn and Co in Slab-Based Nanocomposites for Hybrid Supercapacitors: Impact of Restacking Process on Electrochemical Properties[END_REF]). At the opposite, RMC1b (Figure 3b) showed a dense bulk structure, i.e. a closed porosity that explains its low specific BET surface area (2 m 2 /g) due to the slow acidification step used to flocculate the Mn-and Co-nano-sheets. Regarding the nano-Auger analysis, Auger survey spectra showed O KLL, Mn LMM and Co LMM transitions at about 500 and 480 eV, at about 630, 580 and 530 eV and at about 775, 710 and 645 eV, respectively. SAM overlay images (i.e. elemental 2D distribution from Mn LMM in blue and Co LMM in green) for RMC1a and RMC1b (Figure 3a and 3b, respectively) suggest a nanoscale distribution of Mn / Co. This was further confirmed by Auger survey spectra that always showed Mn and Co LMM transitions simultaneously, i.e. within the 20 nm wide and 2-3 nm depth probe size. Thus, for RMC1a and RMC1b, the restacking procedure occurred at the nanoscale, i.e. the Mn / Co distribution scale is lower than 20 nm. At the opposite, the SAM overlay image for RMC2 (Figure 3c) clearly suggest a microscale distribution of Mn / Co, which is confirmed by Auger survey spectra showing either Mn or Co LMM transitions, which highlights the importance of the restacking procedure. Indeed, contrary to RMC1a and RMC1b, the acidification for RMC2 was performed before mixing the colloidal suspensions together thus favoring the flocculation before an intimate mixture could be obtained. As a result, RMC1a and RMC1b (Figure 3a and3b, respectively) are nano-composites while RMC2 (Figure 3c) is a micro-composite.

Overall, the cross-section SEM/nano-Auger analysis is of high interest to investigate the bulk structure (porosity) together with the bulk morphological organization (distribution scale) of nanocomposites powders with micrometric particles size, which remained a challenge so far. At this point, note that higher magnification could be used to get further insight about the nanoscale distribution but this remains limited by the lack of topographic contrast for such flat samples. An alternative would be to use backscattered electrons to get a Z (atomic number) contrast between elements thus enabling higher magnification/resolution but this mode is not present in our Auger system so far. 4a, the SEM image of the TiSnSb electrode cross-section before cycling showed dense and shapeless TiSnSb particles of 0.5-5 µm surrounded by large and porous areas containing the carbon additives and the CMC binder as confirmed by SAM images and Auger survey spectra. After the 1 st lithiation at 25°C (Figure 4b), TiSnSb particles remained dense with a relatively homogeneous lithiation over the entire electrode thickness as observed from the SAM images. Note that some particles at the bottom were not lithiated more likely because they are not connected to the electronic conductive carbon network of the electrode as observed by SEM (Figure 4b). After 400 cycles at 25°C (Figure 4c), the morphology of the TiSnSb particles significantly changed with the formation of non-uniform porous shell surrounding dense core as observed from both SEM and SAM images. Particles cracks located in the dense cores were also observed (SEM image, Figure 4c). This result thus suggests a shell to core spreading/pulverization of TiSnSb particles over long term cycling. This TiSnSb particles volume expansion phenomenon led to a filling of the electrode porosity as well as to an increase of the electrode thickness, from 12 ±2 µm for the electrode before cycling (Figure 4a) to 15 ±2 µm after the 1 st lithiation (Figure 4b) and finally to 50 ±3 µm after 400 cycles (Figure 4c). Importantly, the Ti, Sn and Sb distribution at the nanoscale (i.e. within the 30 nm wide and 2-3 nm depth resolution for the SAM images) remained relatively homogeneous with only small discrepancies located on the porous shells and despite the huge volume expansion (i.e. huge TiSnSb particles spreading) over long term cycling. This result thus proves that the conversion reaction of TiSnSb into Li3Sb/Li7Sn2/Ti still occurs even after 400 cycles. It also proves that the electrochemically inactive Ti element still participate to the reaction. Finally, after 400 cycles (Figure 4c), fluorine and oxygen elements are detected due to the formation of F-and O-containing SEI species formed by the electrolyte degradation. Interestingly, the SAM images of C, O, F KLL and Ti LMM as well as SAM overlay images of C/O/F and C/O/Ti showed that oxygen and fluorine are located in the porous TiSnSb areas while they are absent from the carbon additive areas. This result thus points to an SEI formation inside the TiSnSb porous shell, in other words as the TiSnSb particles continuously pulverize/break up to form fresh surface over long term cycling.

TiSnSb-based electrodes -Figure

Overall, the cross-section SEM/nano-Auger analysis is of high interest to investigate both the morphological evolution and reaction of conversion materials together with the observation of the SEI formation for micrometer thick electrode and over long term cycling, which remained a challenge so far. At this point, note that one advantage of this approach is the possibility to detect and map the lithium element distribution contrary to most of the analytical techniques including surface sensitive ones. Of course, this remains true only if no other elements overlap with the Li KVV transition such as for Fe. PEO-based solid-state battery stack -It is reminded (as described in the experimental part) that one NMC/PEO-LiTFSI/graphite stack cross-section was typically obtained without optimized Ar + milling parameters (Figure 5a) while the other one was obtained with optimized Ar + milling parameters (Figure 5b). Without optimization (Figure 5a), the SEM image showed that a rough surface was obtained with polymer flowing over the entire cross-section and delamination at the NMC electrode top surface. Such phenomena were partially expected as constraints produced by the Ar + beam can easily damage polymers due to heat dissipation. Therefore, various Ar + beam conditions were tested, i.e. ion beam voltage between 6-16 kV with 2 keV step, ion beam current between 2-4 mA (i.e. 300 µA emission current) with 0.5 step and for different times but always under liquid nitrogen cooling (133 K). Once Ar + milling parameters were optimized (8 keV, 3.5 mA, 6h at 133 K), the SEM image (Figure 5b) showed that a perfectly flat cross-section surface was obtained without apparent physical damage such as polymer flowing. Indeed, the morphology of the different layers and the different interfaces were well defined while the thickness of the layers was preserved. This is supported by SAM images (Figure 5b) of Al KLL (current collector), Co LMM (NMC), S KLL (LiTFSI) and F KLL (LiTFSI + PVDF) showing well defined areas with no spreading/flowing. Interestingly, SAM images of S and F KLL showed that the PEO-LiTFSI electrolyte filled the electrode porosity over the entire electrode thickness. This result highlights that the electrode infusion method by the PEO-LiTFSI/ACN gel followed by the gradual evaporation of ACN is very powerful even for such 50 μm thick electrodes. Finally, typical Auger spectrum (Figure 5b) of an NMC particle taken after Ar + cross-sectioning showed that no chemical change was observed compared to the spectrum of the corresponding pristine NMC powder, i.e. the line shapes matched perfectly and no new transition appeared. Note that this was further confirmed by X-ray spectroscopy analysis (not shown here).

Overall, the Ar + milling cross-section approach is of very high interest to reveal at once all interfaces buried in polymer-based solid-state batteries without physical/chemical damage and despite the high sensitivity of the polymers, which remained a challenge so far. As a matter of fact, to our knowledge, it is the first time this approach is proposed/applied in the literature. At this point, the main drawback of this approach is the possible need for Ar + milling parameters optimization as function of the battery stack system. However, in our case, it was possible to crosssection other battery stacks based on different polymer or gel electrolytes using the optimized ion beam voltage/current while changing only the time due to different stacks thicknesses.

Conclusions

In this work, cross-section by Ar + ion milling is proposed as a powerful method used to access the bulk of composites without physical/chemical change. The sample preparation conditions prior to the Ar + milling cross-sectioning are thus fully described as function of the composites. Moreover, nano-Auger/SEM analysis is proposed to investigate bulk chemical/morphological properties at both micro-and nano-scales for different energy storage composites, namely MnCo-based powders, TiSnSb-based electrodes and PEO-based solid-state battery stacks. For MnCo powders, it allowed revealing the bulk porosity together with the bulk nano-or micro-Mn/Co distribution for micrometric particles. For TiSnSb electrodes, it allowed proving the conversion reaction (and thus the participation of the electrochemically inactive Ti) while revealing the TiSnSb particles morphological evolution (shell to core spreading/pulverization into porous structure) and SEI formation inside the porous TiSnSb, all for micrometer thick electrode over long term cycling. For PEO-based stacks, the cross-section preparation allowed revealing well-defined buried interfaces, which will then be used to follow the interfaces evolution over cycling. Finally, the main advantage and limitation of this cross-section nano-Auger/SEM approach are discussed for each composites. Interestingly, note that the last generation of Auger apparatus has the capacity to perform in situ focused ion beam (FIB) cross-sectioning, which should further increase the interest of this method with minimum sample preparation.

Overall, this work thus opens the door for future development of both Ar + milling cross-section preparation and Auger analysis as powerful tools to reveal and study buried chemical/morphological properties at both micro-and nano-scales in the field of energy storage and beyond, which remained a challenge so far.

Figure 2 .

 2 Figure 2. (top) Schematic representation of the specific sample preparation followed prior to the Ar + milling cross-sectioning as function of the composites and (bottom) resulting typical low magnification SEM images showing the planar surface obtained by the Ar + milling crosssectioning. The white/grey flakes observed in SEM images correspond to the silver epoxy resin.

Figure 3 .

 3 Figure 3. SEM images with SAM overlay (elemental 2D distribution from Mn LMM in blue and Co LMM in green) and Auger spectra taken at indicated positions for the different MnCo composite powders, namely, a) RMC1a, b) RMC1b and c) RMC2. The corresponding exfoliation/restacking procedures are also summarized for clarity.

Figure 4 .

 4 Figure 4. SEM images of TiSnSb electrodes cross-section a) before cycling, b) after the 1 st lithiation at 25°C and c) after 400 cycles at 25°C with selected SAM (elemental 2D distribution) images from C KLL, O KLL, F KLL, Li KVV, Ti LMM, Sn MNN, Sb MNN transitions and some overlays. The area at which the SAM images were taken is indicated in the SEM images. For the TiSnSb electrodes cross-section before cycling (a), Auger spectra are also presented as taken at different position as indicated in the SEM image.

Figure 5 .

 5 Figure 5. SEM images of NMC/PEO-LiTFSI/graphite stack cross-sections a) without and b) with optimized Ar + milling parameters. For the optimized cross-section (b), selected SAM (elemental 2D distribution) images taken at the indicated position from Al KLL, Co LMM, S KLL and F KLL are also presented together with a typical Auger spectrum of an NMC particle compared to a reference spectrum of the corresponding pristine NMC powder.

Acknowledgment

This research was performed in the framework of "Réseau sur le Stockage Electrochimique de l'Energie" (RS2E) and the ANR program no. ANR-10-LABX-76-01. The PEO-based solid-state battery part was carried out as part of the HUB RAISE2024 project funded by Excellence Initiative of Université de Pau et des Pays de l'Adour, I-Site E2S, a French "Investissements d'Avenir" programme.