domain: spi phys info math

text:
N Bérend
email: nicolas.berend@onera.fr

A variant of the Kármán line theory facilitating the identification of a precise altitude boundary

Keywords: Kármán line, spaceflight, atmospheric flight, aeronautics, astronautics

The concept of Kármán Line was first proposed by Andrew G. Haley in a 1957 IAC space law paper, following Kármán's idea to rely on flight mechanics considerations to identify a boundary between atmospheric flight and space flight. The physical phenomenon highlighted by Kármán, and illustrated by Masson & Gazley's diagram, is the fact that, starting from a certain altitude domain, a near-orbital velocity is required to meet equilibrium glide conditions, in which case weight compensation is nearly completely ensured by centrifugal force and lift becomes negligible.

The Kármán line altitude was eventually set by FAI to 100 km in 1960, for the purpose separating aeronautics and astronautics records. Although we know that this altitude was set in accordance with Kármán's idea, it is at least partly an arbitrary choice, as Kármán's guidelines alone are insufficient to define a very precise altitude. Indeed, in a 1961 conference (i.e. one year after he contributed to FAI's choice), Kármán confirms that his proposal for the boundary of space is based on the Masson & Gazley diagram and the phenomenon of centrifugal force being "dominant" (consistently with Haley's and FAI's explanations) but he only points towards the approximate altitude range where this phenomenon is visible (300,000-400,000 ft, or 91-122 km) on the "lift barrier" part of the diagram, rather than a very precise altitude.

In this paper, we propose a new solution to identify a unique altitude threshold while being consistent with Kármán's guidelines. The proposed solution is based on the fact, apparently never mentioned before in literature about the Kármán line, that the equilibrium velocity actually goes through a maximum when altitude increases. The corresponding altitude de facto separates two regions: an atmospheric flight region, where lifting flight behaviour prevails (equilibrium velocity increases with altitude), and a spaceflight region, where orbital flight behaviour prevails (equilibrium velocity decreases with altitude).

We demonstrate, both numerically and theoretically, that this altitude boundary is relatively stable with regards to the vehicle's lift performance, and that it necessary yields near orbital velocity conditions (which fits Kármán's idea). Numerically, this variant of the Kármán line definition is 110 km, which is relatively close to FAI's setting, but with the advantage of being precisely set through flight mechanics considerations. Legal aspects of the boundary of space definition(s) are outside the scope of this paper, which focuses only on flight mechanics/astrodynamics aspects.

Introduction

The Kármán line is one convention for defining the "boundary of space". It is based on Kármán's idea to define such a boundary through flight mechanics considerations, involving the respective roles of centrifugal force and lift in the sustained flight of a lifting vehicle when altitude varies [START_REF] Th | From Low Speed Aerodynamics to Astronautics[END_REF]. Other definitions or proposals for the boundary of space exist, based on different ideas, like the USAF/NASA altitude criterion for "astronaut wings", apparently based on aerodynamic controllability considerations [START_REF] Jenkins | A Word about the definition of space[END_REF], and McDowell's recent proposal, based on considerations of drag and orbit stability for satellites [START_REF] Mcdowell | The edge of space: Revisiting the Kármán Line[END_REF].

The Kármán line concept emerged in space law discussions in the 1950's. The name was coined by Andrew G. Haley space law pioneer, in a 1957 IAC paper [START_REF] Haley | Space Law-The Development of Jurisdictional Concept[END_REF], in which he proposed to define a "jurisdictional boundary" between atmospheric flight and spaceflight, based on Kármán's scientific guidelines.

The Kármán line was adopted in 1960 by FAI (Fédération Aéronautique Internationale) as the boundary separating aeronautic flight records and spaceflight records, and set to 100 km by a group of scientists led by Kármán [START_REF] Sanz Fernández De Córdoba | 100km Altitude Boundary for Astronautics[END_REF]. To this date, this is the only application for which the Kármán line is recognized at international level.

In section 2, we detail the phenomenon highlighted by of Kármán and we provide the equation of the "lift barrier" curve he uses to illustrate his point. We also remind Kármán's own view about the altitude of the boundary of space, according to the most direct source, which is a conference he made in 1961 (i.e. after the choice of the 100 km altitude boundary by FAI).

IAC-22-C1.IP. [START_REF] Sänger | Rocket Flight Engineering (Translation of Translation of "Raketenflugtechnik[END_REF] Page 2 of 9

In this conference, Kármán proposes an altitude range, rather than a precise altitude [START_REF] Th | From Low Speed Aerodynamics to Astronautics[END_REF].

In section 3, we propose an addition to the Kármán line theory, based on a previously unexploited feature of the "lift barrier" curve, which allows identifying more precisely an altitude boundary, based only on flight mechanics considerations.

The original Kármán line definition

General principle and Kármán's guidelines

Kármán's idea to identify a boundary of space is based on a graphics by Masson & Gazley [1][4].

This diagram (Figure 1 from [START_REF] Masson | Surface Protection And Cooling Systems For High-Speed Flight[END_REF]) shows the "corridor of continuous flight" which is the accessible flight domain, represented in altitude vs. velocity coordinates considering technological constraints in aerodynamics (upper curve) and materials (lower curve). [START_REF] Masson | Surface Protection And Cooling Systems For High-Speed Flight[END_REF] (reproduced in Kármán's papers [START_REF] Th | From Low Speed Aerodynamics to Astronautics[END_REF] and [START_REF] Kármán | Aerodynamic Heating-The Temperature Barrier In Aeronautics[END_REF])

The lower curve corresponds to a temperature limitation of the materials. It is usually called the "temperature barrier" [START_REF] Fritz | The Heat Barrier[END_REF] or "heat barrier" [START_REF] Kármán | Aerodynamic Heating-The Temperature Barrier In Aeronautics[END_REF]. It shows, for a given altitude, the maximum velocity a vehicle can fly without exceeding the materials temperature limit.

The upper curve, which is the one that is relevant concerning Kármán's proposition, is the "lift barrier" [START_REF] Fritz | The Heat Barrier[END_REF]. It shows, for a given altitude, the minimum velocity that is required to allow a stable horizontal flight for a lifting vehicle. For convenience, this velocity will be called the "equilibrium velocity" in the following, a short designation reminding that it corresponds to equilibrium flight, i.e. the balance between weight and the sum of lift and centrifugal forces. As will be detailed later, the "lift barrier" curve depends on a hypothesis on the maximum lifting performance that is technologically achievable.

This representation of the "corridor of flight" diagram actually predates Masson & Gazley's paper, which references a 1953 Fritz Haber paper as the source of the graphics [START_REF] Fritz | The Heat Barrier[END_REF].

While F. Haber's paper, in turn, provides no references to other sources, the "lift barrier" curve appears to be much older. The earliest document we could identify showing this curve is a 1933 book by Eugen Sänger [START_REF] Sänger | Rocket Flight Engineering (Translation of Translation of "Raketenflugtechnik[END_REF]. Like Kármán, both Haber and Sänger highlight the fact that the relative contributions of lift and centrifugal forces to weight compensation change when we move along the curve [START_REF] Fritz | The Heat Barrier[END_REF] [START_REF] Sänger | Rocket Flight Engineering (Translation of Translation of "Raketenflugtechnik[END_REF]. Kármán brought the idea of defining an altitude boundary of spaceflight based on this phenomenon.

In [START_REF] Th | From Low Speed Aerodynamics to Astronautics[END_REF], Kármán explains: Gazley considered that continuous flight is limited by two extreme conditions. One condition is that the vehicle is too slow to fly because the weight of the vehicle is larger than the sum of the lift force and centrifugal force (…). It is evident that when the centrifugal force becomes dominant, a vehicle is no longer an airplane but a satellite. It is this point which may be considered the beginning of space. Hence I propose that space begins at 300 or 400,000 ft [91.44 or 121.92 km].

As Figure 1 shows, the 300,000-400,000 ft (~91-122 km) altitude range corresponds to the domain where the lift barrier curve gets suddenly very steep while velocity gets very close to orbital velocity, and tends asymptotically towards it when altitude increases. As will be shown analytically later, this phenomenon also means that the contribution of centrifugal force to the compensation of weight gets very close to 100 %, while the contribution of lift gets close to 0 % (but never gets strictly null, as long as there remains a residual atmospheric density).

We assume that reference [START_REF] Th | From Low Speed Aerodynamics to Astronautics[END_REF], cited above, is the most reliable source about Kármán's own view on the subject of the boundary of space. Indeed, there exist very few original documents explaining Kármán's view on this subject and nearly all of them are indirect sources, not authored by Kármán himself. Reference [START_REF] Th | From Low Speed Aerodynamics to Astronautics[END_REF] is actually the only source we could identify that is undoubtedly from Kármán himself (and alone). It is the transcript of one of Kármán's latest conferences, which, interestingly, occurred in 1961, i.e. one year after the choice of the 100 km altitude limit by FAI.

The other "most direct" sources are: -Haley's papers ( [START_REF] Haley | Space Law-The Development of Jurisdictional Concept[END_REF] and later), which contains some quotes of Kármán, but are not co-authored by him. Haley mentions the altitude of 275,000 ft (83.82 km), but he precise that is condition is "illustrative" of the Kármán line and "may be changed in the future".

IAC-22-C1.IP.8 Page 3 of 9 -Kármán's autobiography [START_REF] Th | The Wind and Beyond: Theodore von Kármán-Pioneer in Aviation and Pathfinder in Space[END_REF], which was posthumously completed by Lee Edson "using the rough material left by the scientist" and published in 1967. The section about the boundary of space appears to be made of excerpts from Haley's papers, save for the fact that the altitude mentioned is 300,000 ft (91.44 km). -FAI's 2004 web article [START_REF] Jenkins | A Word about the definition of space[END_REF] about the Kármán line, written more than 40 years after the setting of the 100 km boundary.

While all these sources consistently mention the role of centrifugal force and lift in the Kármán line theory, they do not explain clearly how this actually leads to the mentioned altitude boundary.

Haley's 275,000 ft altitude (~84 km) is sometimes presented as the original altitude calculated by Kármán [11][12]. This appears to be a misconception, originating from some ambiguity in Haley's papers about what parts of the proposition are from Kármán himself, and also from the fact that indirect sources usually omit that this altitude is presented as "illustrative". In fact, Haley's altitude is not in the domain proposed by Kármán. It is located where the "lift barrier" curve starts only to be steep, while Kármán propose a higher altitude range, where the curve is nearly vertical.

On the other hand, the altitude mentioned in Kármán's posthumous autobiography (300,000 ft or ~91 km), is partly consistent with the altitude range from [START_REF] Th | From Low Speed Aerodynamics to Astronautics[END_REF], as it is its lower bound. A possible explanation is that the biography's co-author may have chosen this figure as the best compromise between Haley's papers (from which some sentences are reproduced identically) and Kármán's conference.

Finally, FAI's altitude (100 km) is definitely within the altitude domain proposed by Kármán in [START_REF] Th | From Low Speed Aerodynamics to Astronautics[END_REF], and we know that it was retained partly because it was a round figure, so this is consistent.

Therefore, according to reference [START_REF] Th | From Low Speed Aerodynamics to Astronautics[END_REF], and unless previously unknown documents from Kármán surface, it appears that, from the purely scientific point of view, Kármán did not identify a precise altitude for the boundary of space but rather a rough altitude range. This range corresponds to domain where the "lift barrier" curve gets very steep, which is a visual sign that the weight is nearly entirely balanced by centrifugal force and lift is nearly null.

Analytical study of the "lift barrier curve"

As explained in the legends of Masson & Gazley's figure (Figure 1), in Kármán explanations and also in other papers reproducing the same figure [START_REF] Hermann | Problems of Hypersonic Flight at the Re-Entry of Satellite Vehicles, IX th International Copyright[END_REF] [START_REF] Hermann | Re-Entry Trajectories and Problems of Hypersonic Flow, Space Trajectory[END_REF], the equation that describes the upper limit of the "corridor of continuous flight" is the equation of the balance between weight and the sum of lift and centrifugal force:

Weight = Lift + Centrifugal force Eq. [ 1 ]
Or, in terms of acceleration:

𝑔(ℎ) = 1 2 𝜌(ℎ)𝑉 2 ( 𝑀 𝑆 • 𝐶 𝐿 ) + ( 𝑉 2 𝑟 𝐸 + ℎ )
Eq. [START_REF] Jenkins | A Word about the definition of space[END_REF] with:

h= altitude g(h)= gravity acceleration at altitude h (h)= atmospheric density at altitude h V = velocity rE = Earth radius M = mass (kg). CL = lift coefficient (-) S = aerodynamic surface (m 2 ).
M/(S.CL) is the "lift parameter" [START_REF] Hirschel | Selected Aerothermodynamic Design Problems of Hypersonics Flight Vehicles[END_REF] expressed in kg/m 2 . This coefficient is the link between the ambient dynamic pressure to the lift acceleration, so it characterizes the lift performance of the vehicle. The lower this coefficient is, the better the lift performance is. Therefore, reducing the lift parameter makes the lift barrier curve higher on the velocity/altitude diagram, which enlarges the "corridor of continuous flight". In the context of the "lift barrier" the lift parameter should represent the technological limit.

The lift parameter can be expressed as a function of other key design parameters. Indeed, it is ratio between the wing loading (M/S) and the lift coefficient (CL), and also between the ballistic coefficient M/(S.CD) and the lift-to-drag ratio (CL/CD):

𝑀 𝑆 • 𝐶 𝐿 = ( 𝑀 𝑆 ) 𝐶 𝐿 = ( 𝑀 𝑆 • 𝐶 𝐷 ) ( 𝐶 𝐿 𝐶 𝐷 )
Eq. [START_REF] Mcdowell | The edge of space: Revisiting the Kármán Line[END_REF] The most direct way to plot the lift barrier curve in the velocity/altitude reference frame is to extract velocity from Eq. [START_REF] Jenkins | A Word about the definition of space[END_REF]:

𝑉 = √ (𝑟 𝐸 + ℎ) • 𝑔(ℎ) 1 + (𝑟 𝐸 + ℎ) • 𝜌(ℎ) 2 ( 𝑀 𝑆 • 𝐶 𝐿 )
Eq. [START_REF] Haley | Space Law-The Development of Jurisdictional Concept[END_REF] Alternatively, another way to plot the curve is to set velocity and to solve Eq. [START_REF] Jenkins | A Word about the definition of space[END_REF] for the altitude, but it would need to be done numerically, as there is no closed-form solution.

Velocity V, as expressed by Eq. [START_REF] Haley | Space Law-The Development of Jurisdictional Concept[END_REF], is the equilibrium velocity at altitude h, i.e. the exact velocity that is required for a stable, locally horizontal flight. IAC-22-C1.IP.8

Page 4 of 9 Indeed, Eq. [START_REF] Jenkins | A Word about the definition of space[END_REF] is a particular case of the state dynamics equation of the flight path angle () [START_REF] Hirschel | Selected Aerothermodynamic Design Problems of Hypersonics Flight Vehicles[END_REF] for =0°:

𝑉 • 𝛾̇= cos 𝛾 [( 𝑉 2 𝑟 𝐸 + ℎ ) -𝑔(ℎ)] + 1 2 𝜌(ℎ)𝑉 2 ( 𝑀 𝑆 • 𝐶 𝐿 )
Eq. [START_REF] Sanz Fernández De Córdoba | 100km Altitude Boundary for Astronautics[END_REF] As precised in Masson & Gazley's paper [START_REF] Masson | Surface Protection And Cooling Systems For High-Speed Flight[END_REF] (and also in the Fritz Haber paper that is referenced as the source of the graphics [START_REF] Fritz | The Heat Barrier[END_REF]), the "lift barrier" curve is plotted considering a 80 psf lift parameter, which is around 400 kg/m 2 (390.6 kg/m 2 ). This value is "a very low figure by today's standards", wrote Fritz Haber in this 1953 article [START_REF] Fritz | The Heat Barrier[END_REF].

Figure 2 allows to verify that Masson & Gazley's lift barrier curve (red curve, which is digitized from the version appearing in Haley's paper [START_REF] Haley | Space Law-The Development of Jurisdictional Concept[END_REF]) indeed coincides with the curve described by Eq. [START_REF] Haley | Space Law-The Development of Jurisdictional Concept[END_REF] and a 80 psf lift parameter (dark blue curve).

Figure 2 -Re-creation of the "lift barrier" curve using

Eq. [START_REF] Fritz | The Heat Barrier[END_REF] The 80 psf curve has been re-calculated considering the following models:

-The atmospheric density model ((h)) from the ARDC 1956 model [START_REF] Minzner | The ARDC atmosphere model[END_REF], which was the standard at that time. -A simple Kepler gravity acceleration model (g(h)), defined as follows:

𝑔(ℎ) = 𝑔 0 ( 𝑟 𝐸 𝑟 𝐸 + ℎ ) 2
Eq. [START_REF] Masson | Surface Protection And Cooling Systems For High-Speed Flight[END_REF] where g0 and rE are set consistently with the atmosphere model [START_REF] Minzner | The ARDC atmosphere model[END_REF]: g0= 9.80665 m/s 2 = sea-level gravity acceleration at latitude 45°32'40" rE = 6356.755 m = effective Earth radius at latitude 45°32'40" Using Eq. [START_REF] Masson | Surface Protection And Cooling Systems For High-Speed Flight[END_REF], Eq. [START_REF] Haley | Space Law-The Development of Jurisdictional Concept[END_REF] becomes:

𝑉 = √ 𝑟 𝐸 • 𝑔 0 (1 + ℎ 𝑟 𝐸 ) • (1 + (𝑟 𝐸 + ℎ) • 𝜌(ℎ) 2 ( 𝑀 𝑆 • 𝐶 𝐿 ) )
Eq. [START_REF] Fritz | The Heat Barrier[END_REF] Figure 2 also illustrates two phenomena that occurs in the low velocity and high velocity domain, respectively:

 At low velocity: the lift barrier curve tends to be the same as the constant 80 psf dynamic pressure curve (light blue dotted curve), which illustrates the fact that in this domain, the weight is balanced nearly entirely by lift, while the centrifugal force is negligible.  Conversely, at high velocity: the lift barrier curve tends to be the same as the orbital velocity curve (green), which illustrates the fact the weight is balanced nearly entirely by centrifugal force lift, while the contribution lift is negligible.

The link between the lift barrier curve and the phenomenon of "centrifugal force taking over the lift" appears more clearly by introducing the ratio between centrifugal force and weight, that will be denoted k:

𝑘 = ( 𝑉 2 𝑟 𝐸 + ℎ ) 𝑔(ℎ)
Eq. [START_REF] Sänger | Rocket Flight Engineering (Translation of Translation of "Raketenflugtechnik[END_REF] Since orbital velocity at altitude h is:

𝑉 𝑜𝑟𝑏 = √(𝑟 𝐸 + ℎ) • 𝑔(ℎ)
Eq. [START_REF] Kármán | Aerodynamic Heating-The Temperature Barrier In Aeronautics[END_REF] k can be expressed as a function of the ratio between velocity and orbital velocity:

𝑘 = ( 𝑉 𝑉 𝑜𝑟𝑏 ) 2
Eq. [START_REF] Th | The Wind and Beyond: Theodore von Kármán-Pioneer in Aviation and Pathfinder in Space[END_REF] Eq. [START_REF] Th | The Wind and Beyond: Theodore von Kármán-Pioneer in Aviation and Pathfinder in Space[END_REF] illustrates the fact that, when velocity tends towards orbital speed, the contribution of centrifugal force to the compensation of weight tends towards 100 % (while the contribution of lift tends towards 0 %).

Transforming Eq. [ 4 ] using Eq. [START_REF] Kármán | Aerodynamic Heating-The Temperature Barrier In Aeronautics[END_REF], we get:

𝑉 𝑉 𝑜𝑟𝑏 = √ 1 1 + (𝑟 𝐸 + ℎ) • 𝜌(ℎ) 2 ( 𝑀 𝑆 • 𝐶 𝐿 )
Eq. [START_REF] Goedhart | The never ending dispute : delimitation of air space and outer space[END_REF] Then, with Eq. [START_REF] Th | The Wind and Beyond: Theodore von Kármán-Pioneer in Aviation and Pathfinder in Space[END_REF]:

𝑘 = 1 1 + (𝑟 𝐸 + ℎ) • 𝜌(ℎ) 2 ( 𝑀 𝑆 • 𝐶 𝐿 )
Eq. [START_REF] Th | The Non Kármán Line: An Urban-Legend of the Space Age[END_REF] Copyright 2022 by ONERA -The French Aerospace Lab. Published by the IAF, with permission and released to the IAF to publish in all forms IAC-22-C1.IP.8 Page 5 of 9

Eq. [START_REF] Th | The Non Kármán Line: An Urban-Legend of the Space Age[END_REF] shows that the ratio between centrifugal force and weight increases with the altitude (i.e. when we move upward along the "lift barrier curve") and tends towards 100 % (while the contribution of lift tends towards 0 %) : this is the precise phenomenon highlighted by Kármán (and also, as noted earlier, by other authors [7][8]).

A detailed study of the variation of the altitude boundary as a function of k and the lift parameter, is given in a previous paper [START_REF] Bérend | The Missing Calculation behind the Original Kármán Line Definition -A Credible Hypothesis[END_REF].

It is important to note that k (then V/Vorb) never gets strictly equal to 100 % along the lift barrier curve, as long as we consider that atmospheric density never gets strictly null. Since we cannot make this assumption (which would have answered the question of the boundary of space, in the first place !), the hypothesis behind the lift barrier curve excludes the possibility of reaching exactly the orbital velocity or compensating 100 % of the weight using centrifugal force.

With this respect, Haley's paper [START_REF] Haley | Space Law-The Development of Jurisdictional Concept[END_REF] and FAI's article [START_REF] Sanz Fernández De Córdoba | 100km Altitude Boundary for Astronautics[END_REF] appear to be inexact, which seems to be the origin of a common misconception about the Kármán line. Indeed, they associate the crossing of the Kármán line to the vanishing of lift ("Lift is gone" [START_REF] Haley | Space Law-The Development of Jurisdictional Concept[END_REF]) and a velocity reaching or even exceeding the orbital velocity ("[The speed required for lift] could be close or even bigger than the circular orbital speed at that altitude" [START_REF] Sanz Fernández De Córdoba | 100km Altitude Boundary for Astronautics[END_REF]). The contradiction is actually obvious in Haley's paper, as the "illustrative" conditions mentioned are a velocity of 25,000 ft/s (7620 m/s) and an altitude of 275,000 ft (~84 km): these are suborbital conditions (suborbital velocity, hence requiring a contribution of lift to ensure a sustained horizontal flight). The many contradictions in the early literature about the Kármán line altitude have been highlighted by Gangale in [START_REF] Th | The Non Kármán Line: An Urban-Legend of the Space Age[END_REF], who suggests that "von Kármán himself never proposed a boundary at 100 kilometers" and that "reference to the100-kilometer line as the von Kármán line is a misnomer" and an "urban legend of the space age". Our findings (namely reference [START_REF] Th | From Low Speed Aerodynamics to Astronautics[END_REF]) are partly in line with the first part of these conclusions, as they tend to confirm that the 100 km is notby itselfthe result of a precise calculation by Kármán. But reference [START_REF] Th | From Low Speed Aerodynamics to Astronautics[END_REF] also proves than Kármán himself undoubtedly proposed an altitude range based on the same flight mechanics phenomenon (and graphics) as described in Haley's paper [START_REF] Haley | Space Law-The Development of Jurisdictional Concept[END_REF], and this range is definitely consistent with the 100 km altitude, which, as we know, was eventually set to a round number ("apparently [by] Von Karman himself", according to [START_REF] Sanz Fernández De Córdoba | 100km Altitude Boundary for Astronautics[END_REF]). Conversely, the 84 km/275,000 ft altitude should not be attributed to Kármán, as this was no more than an "illustrative" figure [START_REF] Haley | Space Law-The Development of Jurisdictional Concept[END_REF] provided by Haley (a lawyer), a precision that is, unfortunately, often omitted. Incidentally, when we identify these facts, the mention by Haley in [START_REF] Haley | Survey of Legal Opinion on Extraterrestrial Jurisdiction[END_REF] that "[FAI's 100 km altitude] coincides with the Kármán theory" appears no longer as a contradiction compared to his earlier paper [START_REF] Haley | Space Law-The Development of Jurisdictional Concept[END_REF]. It is rather a further confirmation that Kármán's work is indeed at the origin of the 100 km limit, even though the eventual choice of a round number is partly arbitrary. The purpose of the variant of the Kármán line definition that will be presented in the next section, is, precisely, to avoid resorting to an arbitrary choice to get a unique altitude.

Proposed variant of the Kármán line definition

The "Maximum Equilibrium Velocity (MEV) altitude"

The "lift barrier" curve on Masson & Gazley's diagram (Figure 1) appears to be simplified compared to the actual curve obtained using Eq. [ 4 ] (Figure 2). Indeed, in the high velocity domain, it shows a vertical asymptote, with the altitude continuously increasing when velocity increases. This is actually a simplified representation of the asymptote, which, as noted in the previous section, corresponds to orbital velocity (green curve on Figure 2).

The orbital velocity curve is not vertical but slightly titled, since orbital velocity decreases as altitude increases. Conversely, when we move along the lift barrier towards high altitudes, velocity gets closer to orbital velocity but it actually decreases in absolute value, while it is increasing with altitude in the lower altitude domain of the graphics.

This can be observed more clearly on Figure 3, which is a zoomed version of Figure 2 around the domain where this phenomenon occurs. As a consequence (and it will be later demonstrated analytically using a simplified atmospheric density model), the equilibrium velocity goes through a maximum value. This maximum occurs around 110 km (with the same ~400 kg/m 2 lift parameter hypothesis). Copyright 2022 by ONERA -The French Aerospace Lab. Published by the IAF, with permission and released to the IAF to publish in all forms IAC-22-C1.IP.8

Page 6 of 9

Note that this is quite close to the middle of the altitude range identified by Kármán (while the FAI 100 km altitude is comparatively farther).

A corollary of the existence of a Maximum Equilibrium Velocity altitude (now abbreviated as "MEV altitude" for convenience) is that is separates two altitude domains:

-At lower altitudes: a domain where the equilibrium velocity increases with altitude. This property can be seen as an "aeronautical behaviour" (for a lifting vehicle, a higher altitude means a higher velocity). -At higher altitudes: a domain where the equilibrium velocity decreases with altitude. This property may be described as "spaceflight" or "orbital behaviour" (for a space vehicle in a circular orbit, a higher altitudes means a lower velocity).

With the MEV altitude, we now have identified a physical phenomenon that is:

fully consistent with Kármán's ideas (as will be demonstrated later, it necessarily occurs at conditions where centrifugal force is dominant and lift negligible). -occurs at a definite altitude (while Kármán's proposal was a limited to a rough altitude range).

Due do these properties, the MEV altitude can be seen as a variantor, more exactly, an addition -to Kármán's guidelines for the definition of an altitude boundary between atmospheric flight and spaceflight. Following these observations, the following question naturally arises: could this "variant" actually be what Kármán had in mind ? All the elements available tend to show that the answer is "no" and that this particular proposition for the boundary of space definition is indeed new. Indeed, no document in Kármán line literature refers to the particular phenomenon we highlight (equilibrium glide velocity going through a maximum when altitude varies). In particular, this phenomenon is not mentioned at all in the most direct sources we have about the Kármán line theory (by Haley, Kármán or FAI).

All these sources provide only qualitative explanations about the relative roles of centrifugal force and lift, but no precise mathematical or physical definition allowing to identify a precise altitude.

Furthermore, in its 1961 conference, still illustrates his theory using the original "Masson & Gazley" diagram (Figure 1), which shows a simplified version of the "lift barrier" curve (with a vertical asymptote, thus hiding the existence of a MEV altitude). If Kármán had wanted to highlight the concept of MEV altitude, he would had shown a "regular" version of the lift barrier curve on which the MEV is visible and he would have mentioned this phenomenon.

For the sake of completeness, it shall be noted that Haley's version of the diagram (which he says to have "unskilfully redone" [START_REF] Haley | Space Law-The Development of Jurisdictional Concept[END_REF]) actually shows a qualitatively correct bending of the lift barrier asymptote. However, as analysed in a previous paper [START_REF] Bérend | The Missing Calculation behind the Original Kármán Line Definition -A Credible Hypothesis[END_REF], a likely explanation is that Haley (who was not a scientist but a lawyer) has mixed the original Masson & Gazley curves with elements from a similar flight domain diagram by Sänger (also reproduced in the same Haley paper). Beside the graphic itself, Haley's explanations never refers to the existence of an altitude where the equilibrium velocity is minimum, and the "illustrative" altitude/velocity conditions he provides are placed much lower on the graphics.

Sensitivity to the lift parameter

The lift barrier curve, hence the MEV altitude, depends on an estimation of the minimum lift parameter (M/(S.CL)) that is technically achievable.

As explained previously, the original Masson & Gazley figure considered a ~400 kg/m 2 (80 psf) lift parameter. Today, this value is still low compared to today's technology, so it seems to be still relevant as an "optimistic value" representing the technology limitations.

Nevertheless, such an estimation is, by definition, uncertain, so it is interesting to observe how the MEV altitude varies with it.

Figure 4 shows the lift barrier curves for different lift parameters between 100 and 2000 kg/m 2 (which is very large range). For each curve, the MEV point is indicated with a red diamond. It shall be noted than changing the lift parameter hypothesis by a factor 2 (in either ways) induces a relatively small change in the MEV altitude (less than 5 km), compared to its original 110 km altitude. That rate of change of the MEV altitude is around 1.3 km for each 200 kg/m 2 variation.

This relative stability of the MEV altitude with regards to the technological hypothesis make it an interesting candidate for the definition of an altitude boundary based on scientific considerations.

Closed-formed estimation

The MEV altitude (hMEV) is the altitude that maximizes the equilibrium velocity, as expressed by Eq. [START_REF] Fritz | The Heat Barrier[END_REF], so we have:

ℎ 𝑀𝐸𝑉 = argmax ℎ √ 𝑟 𝐸 • 𝑔 0 (1 + ℎ 𝑟 𝐸 ) • (1 + (𝑟 𝐸 + ℎ) • 𝜌(ℎ) 2 ( 𝑀 𝑆 • 𝐶 𝐿 ) )
Eq. [START_REF] Hermann | Problems of Hypersonic Flight at the Re-Entry of Satellite Vehicles, IX th International Copyright[END_REF] An equivalent expression is:

ℎ 𝑀𝐸𝑉 = argmin ℎ [(𝑟 𝐸 + ℎ) • (1 + (𝑟 𝐸 + ℎ) • 𝜌(ℎ) 2 ( 𝑀 𝑆 • 𝐶 𝐿 ) )]
Eq. [START_REF] Hermann | Re-Entry Trajectories and Problems of Hypersonic Flow, Space Trajectory[END_REF] So hMEV satisfies: )] = 0

Eq. [START_REF] Hirschel | Selected Aerothermodynamic Design Problems of Hypersonics Flight Vehicles[END_REF] or:

(1 + 𝜌(ℎ) • (𝑟 𝐸 + ℎ) 2 ( 𝑀 𝑆 • 𝐶 𝐿 ) ) + (𝑟 𝐸 + ℎ) 2 ( 𝑀 𝑆 • 𝐶 𝐿 ) • ( 𝑑𝜌(ℎ) 𝑑ℎ • (𝑟 𝐸 + ℎ) + 𝜌(ℎ)) = 0
Eq. [START_REF] Minzner | The ARDC atmosphere model[END_REF] Let's now consider an exponential atmospheric density model:

𝜌(ℎ) = 𝜌 0 • 𝑒 -ℎ 𝐿
Eq. [START_REF] Bérend | The Missing Calculation behind the Original Kármán Line Definition -A Credible Hypothesis[END_REF] with: 0 = atmospheric density at sea-level L = scale height With this model, Eq. [START_REF] Minzner | The ARDC atmosphere model[END_REF] becomes:

𝑀 𝑆 • 𝐶 𝐿 = 𝜌 0 • 𝑒 -ℎ 𝐿 • (𝑟 𝐸 + ℎ) ( (𝑟 𝐸 + ℎ) 2𝐿 -1)
Eq. [START_REF] Haley | Survey of Legal Opinion on Extraterrestrial Jurisdiction[END_REF] Considering that 𝑟 𝐸 ≫ ℎ and 𝑟 𝐸 ≫ 2𝐿, we get:

𝑀 𝑆 • 𝐶 𝐿 ≅ 𝜌 0 • 𝑒 -ℎ 𝐿 • 𝑟 𝐸 2 2𝐿
Eq. [START_REF] Walter | Astronautics -The Physics of Space Flight[END_REF] Then, solving Eq. [START_REF] Walter | Astronautics -The Physics of Space Flight[END_REF], we get a closed form estimation of the MEV altitude:

ℎ 𝑀𝐸𝑉 ≅ L • ln ( 𝜌 0 • 𝑟 𝐸 2 2𝐿 ( 𝑀 𝑆 • 𝐶 𝐿 ) )
Eq. [START_REF]NASA -Lunar and Planetary Science on NASA Space Science Data Coordinated Archive[END_REF] This formula allows to get easily an estimation of altitude hMEV using only a few parameters:

-A technical parameter: an estimation of the minimum achievable lift parameter (M/(S.CL)). -Three parameters related to the planetoid model (rE) and the atmosphere model (0 and L), respectively.

With the same simplifying assumption, we can now get an estimation of the ratio between centrifugal force to weight ratio at hMEV boundary (kMEV).

From Eq. [START_REF] Walter | Astronautics -The Physics of Space Flight[END_REF], we have:

𝜌(ℎ 𝑀𝐸𝑉 ) ( 𝑀 𝑆 • 𝐶 𝐿 ) ≅ 2𝐿 𝑟 𝐸 2 Eq. [ 21 ]
Which, after replacement in Eq. [START_REF] Th | The Non Kármán Line: An Urban-Legend of the Space Age[END_REF], yields:

𝑘 𝑀𝐸𝑉 ≅ 1 1 + 𝐿 𝑟 𝐸

Eq. [ 22 ]

Considering L = 6.7 km for the exponential atmospheric density for Earth [START_REF] Walter | Astronautics -The Physics of Space Flight[END_REF], we have 𝐿 𝑟 𝐸 ⁄ ≅ 1.05 • 10 -3 . Since 𝐿 ≪ 𝑟 𝐸 , Eq. [ 21 ] shows that kMEV is necessarily close to 1 ( 99.88 %), and, due to Eq. [START_REF] Th | The Wind and Beyond: Theodore von Kármán-Pioneer in Aviation and Pathfinder in Space[END_REF], that 𝑉 𝑉 𝑜𝑟𝑏 ⁄ is even closer to 1 ( 99.95 %). In other words, these closed form expressions demonstrate that the maximum equilibrium velocity is necessarily close to orbital velocity, with a very small (but non zero) contribution of lift. This feature is consistent with Kármán's general guidelines about the region on of the lift barrier curve where the "boundary of space" should be.

Using the same lift parameter hypothesis (390.6 kg/m 2 or 80 psf), we verify that Eq. [START_REF]NASA -Lunar and Planetary Science on NASA Space Science Data Coordinated Archive[END_REF] provides a valid estimation of hMEV: 110.01 km, which is nearly identical to the value we get with the regular atmospheric density model and complete numerical solving (cf. previous section).

Out of curiosity and for the purpose of comparison, we have also computed the estimations of hMEV for Mars and Venus (using the same lift parameter hypothesis).

IAC-22-C1.IP.8

Page 8 of 9

The results for the three planets (as well as the model parameters) are given in Table 1. The sources for the parameter models are [START_REF] Walter | Astronautics -The Physics of Space Flight[END_REF] for Earth and [START_REF]NASA -Lunar and Planetary Science on NASA Space Science Data Coordinated Archive[END_REF] for Mars and Venus.

Planet 0 (kg/m 2 ) rE (km) L (km) hMEV, (km) Surprisingly, the MEV altitude for Mars (113 km) is very close to the one of Earth, while it is much higher for Venus (303 km).

Conclusion

We have proposed a variantor, more exactly an addition -to the original Kármán line theory. It is defined as the altitude the where equilibrium velocity (which is the velocity allowing a locally horizontal flight for a lifting vehicle) is maximum.

It separates two altitude domains that are relevant to the distinction between atmospheric flight and spaceflight: a low altitude domain where "aeronautics behaviour" prevails (the equilibrium velocity increases with altitude) and a high altitude domain where the "spaceflight behaviour" prevails (opposite effect).

The interest of this proposal is twofold:

-It is fully compatible with Kármán's original idea (relying on the "lift barrier curve" and the phenomenon of "centrifugal force taking over") and with the altitude range he proposed (92-122 km). -It provides a method to identify a precise altitude boundary using flight mechanics considerations and a technical hypothesis (minimum achievable lift parameter), while Kármán proposed only a rough altitude interval [START_REF] Th | From Low Speed Aerodynamics to Astronautics[END_REF] and FAI selected arbitrarily a round number (100 km) in the domain pointed by Kármán [START_REF] Sanz Fernández De Córdoba | 100km Altitude Boundary for Astronautics[END_REF].

According to this definition, and using the same lift parameter hypothesis as the original "lift barrier curve" used by Kármán (~400 kg/m 2 ), the altitude for the boundary of space is located around 110 km. This altitude is relatively stable when the lift parameter hypothesis varies. This proposal should be seen primarily as an answer to the scientific question: "is there a physical phenomenon, pertaining to flight mechanics, allowing to identify an altitude boundary between atmospheric flight and space flight ?" While this proposal may contribute to the debates, the question of setting (or adapting) a precise "boundary of space" altitude for particular applications (space law, space records, astronaut wings, etc.) is outside the scope of this paper, as, in any case, it is a matter of convention.

Figure 1 -

 1 Figure 1 -"Masson & Gazley diagram" showing the corridor of continuous flight [6] (reproduced in Kármán's papers [1] and [9])

Figure 3 -

 3 Figure 3 -Zoom on the asymptote of the lift barrier

Figure 4 -

 4 Figure 4 -Zoom on the lift barrier curve for different lift parameter hypothesesIf we consider a factor 2 uncertainty on the ~400 kg/m 2 hypothesis (which seems a reasonable assumption), we get:-For 200 kg/m 2 (= half the original hypothesis): MEV altitude ~ 113 km -For 800 kg/m 2 (= twice the original hypothesis):MEV altitude ~ 105 km)

Table 1 -

 1 Comparison of the MEV altitude for Earth, Mars and Venus (with M/(S.CL)= 390.6 kg/m 2 or 80 psf).

		1.752	6356.766	6.7	110.01
	Mars	0.02	3389.5	11.1	113.05
	Venus	65	6051.8	15.9	303.23

rd International Astronautical Congress (IAC), Paris, France, 18-22 September 2022.Copyright 2022 by ONERA -The French Aerospace Lab. Published by the IAF, with permission and released to the IAF to publish in all forms


Romain Lhermerout

Christophe Diederichs

Susan Perkin
email: susan.perkin@chem.ox.ac.uk

Are Ionic Liquids Good Boundary Lubricants? A Molecular Perspective

Keywords: ionic liquid, boundary lubrication, molecular friction, nanotribology

The application of ionic liquids as lubricants has attracted substantial interest over the past decade and this has produced a rich literature. The aim of this review is to summarize the main findings about frictional behavior of ionic liquids in the boundary lubrication regime. We first recall why the unusual properties of ionic liquids make them very promising lubricants, and the molecular mechanisms at the origin of their lubricating behavior. We then point out the main challenges to be overcome in order to optimise ionic liquid lubricant performance for common applications. We finally discuss their use in the context of electroactive lubrication.

Introduction

Friction is a very general phenomenon that appears as soon as one tries to set two bodies in relative motion [START_REF] Bowden | The Friction and Lubrication of Solids[END_REF][START_REF] Kardar | The "friction" of vacuum, and other fluctuation-induced forces[END_REF]. It is a dissipative process during which a part of the input energy originally injected to produce a global motion is in fact converted into heat. Friction therefore plays a crucial role in many applications, for which the heat production has to be maximized (matches, tyres, brakes), or minimized (turbines, motors), or at least controlled. Adding a liquid between the two solids is a well-known way to influence friction, and the lubrication effect dramatically depends on the nature of the liquid used. In general, real surfaces are rough so friction takes place on many asperities (Figure 1a).

The link between the lubrication mechanisms on a single asperity and the macroscopic motion is complex, because of the possible emergence of collective effects. In the following, we will focus on the so-called boundary lubrication regime, namely when the two solids are separated by a liquid film of a few molecules thickness on a single asperity. For reviews about macroscopic friction across ionic liquids, we refer to [START_REF] Bermúdez | Ionic Liquids as Advanced Lubricant Fluids[END_REF][START_REF] Minami | Ionic Liquids in Tribology[END_REF][START_REF] Palacio | A Review of Ionic Liquids for Green Molecular Lubrication in Nanotechnology[END_REF].

A "good lubricant" could be defined in terms of three principal requirements. (i) The friction coefficient achieved should be controlled. A very low friction, or superlubricity is not necessarily required in applications; instead it is often desirable to have highly controlled friction in order to create very reproducible motion in response to an external force or perturbation. Examples are the MEMS gyroscopes and capacitive accelerometers found in typical mobile phones/devices; (ii) The liquid film should be able to sustain the relevant pressure without being squeezed-out. Typical pressure experienced by boundary films on rough surfaces can indeed be extremely high because the normal load is concentrated on microscopic asperities. The sustained presence of liquid not only maintains lubrication but often also serves to reduce wear; (iii) The lubrication performance should be robust, i.e., minimally dependent on changes in the environment. This last condition is particularly challenging given the multiple perturbations potentially at play (high temperatures, pressures, shear stresses, contamination by moisture, etc.) that can lead to many uncontrolled modifications of the system by complex tribological processes (wear, phase transition, (electro)chemical decomposition, etc.). Ionic liquids are pure salts that are liquid under room temperature-pressure conditions.

They exhibit an exceptional stability (low volatility, nonflammability, wide electrochemical and thermal windows), which makes them very promising systems for lubrication and many other applications (energy storage, catalysis, electrowetting, etc.) [START_REF] Hallett | Room-Temperature Ionic Liquids: Solvents for Synthesis and Catalysis[END_REF][START_REF] Fedorov | Ionic Liquids at Electrified Interfaces[END_REF][START_REF] Hayes | Structure and Nanostructure in Ionic Liquids[END_REF]. Additionally, it has been observed that ionic liquids are very hard to squeeze-out and can produce low friction [START_REF] Perkin | Layering and shear properties of an ionic liquid, 1-ethyl-3-methylimidazolium ethylsulfate, confined to nano-films between mica surfaces[END_REF], irrespective of the temperature [START_REF] Hjalmarsson | Is the boundary layer of an ionic liquid equally lubricating at higher temperature?[END_REF], thus they could potentially be very good lubricants.

In this review, we will summarize the main questions, findings and perspectives in the field of friction across ionic liquids in the boundary lubrication regime. The aims are to discuss the current state of knowledge regarding the molecular-level mechanisms of dissipation across ionic liquid films, and, from this, to identify important molecular considerations when designing ionic liquid lubricants for various purposes. The paper is organized as follows. In Section 2, we will review studies giving insight into the relation between the molecular structure and the frictional behavior. In Section 3, we will focus on how the lubrication performances vary if the ionic liquid is not pure, i.e., with a small fraction of water present or diluted in a solvent. Section 4 will be dedicated to techniques to "switch" friction properties, namely to change friction by varying external parameters and in a reversible way.

The Molecular Mechanisms of Friction Across Ionic Liquids

The Surface Force Balance/Apparatus and the Colloidal Probe Atomic Force Microscope are two commonly used techniques to directly measure friction in the boundary lubrication regime [11].

The principle of these experiment is sketched in Figure 1b. The two solid surfaces are at the same time curved at large scale (radius of curvature R, ca.1 cm in typical Surface Force Balance experiments) and smooth at the molecular scale, to ensure a single contact point. The liquid thickness D is set to a known value with a precision better than the molecular size, one surface is moved laterally at a velocity v and the corresponding normal force F N and lateral force F L are measured.

When confined between two solid surfaces at the nanoscale, an ionic liquid tends to be squeezed-out discontinuously, by discrete steps, arising from the organized arrangement of molecules in ordered layers. This produces a damped oscillation for the force profile, called structural or solvation force (Figure 2a). Such phenomena are not specific to ionic liquids; structural forces are observed with apolar solvents, salt solutions, polymer melts, etc. [START_REF] Horn | Direct measurement of structural forces between two surfaces in a nonpolar liquid[END_REF][START_REF] Christenson | Experimental measurements of solvation forces in nonpolar liquids[END_REF][START_REF] Israelachvili | Molecular layering of water at surfaces and origin of repulsive hydration forces[END_REF][START_REF] Horn | Molecular organization and viscosity of a thin film of molten polymer between two surfaces as probed by force measurements[END_REF]. However, ionic liquid structure consists of alternating layers of excess cation density and excess anion density, and one squeeze-out event corresponds in fact to the squeeze-out of an electroneutral 'slab' which can be considered, approximately, as two layers (one cation-and one anion-excess) together [START_REF] Perkin | Layering and shear properties of an ionic liquid, 1-ethyl-3-methylimidazolium ethylsulfate, confined to nano-films between mica surfaces[END_REF][START_REF] Horn | Double-layer and solvation forces measured in a molten salt and its mixtures with water[END_REF][START_REF] Atkin | Structure in Confined Room-Temperature Ionic Liquids[END_REF][START_REF] Bou-Malham | Nanoconfined ionic liquids: effect of surface charges on flow and molecular layering[END_REF][START_REF] Ueno | Resonance shear measurement of nanoconfined ionic liquids[END_REF][START_REF] Hayes | Double Layer Structure of Ionic Liquids at the Au(111) Electrode Interface: An Atomic Force Microscopy Investigation[END_REF][START_REF] Perkin | Self-assembly in the electrical double layer of ionic liquids[END_REF][START_REF] Smith | Monolayer to Bilayer Structural Transition in Confined Pyrrolidinium-Based Ionic Liquids[END_REF]. In several cases of long-chain and geometrically irregular ionic liquids the squeeze-out behaviour is found to become continuous, rather than discontinuous. The amplitude of the oscillations are in general higher for ionic liquids, arising from the high electrostatic barriers to reorganisation, and the last layer of cations is usually never squeezed-out, even under normal pressures of ~10 MPa. The condition (ii) to have a good lubricant, i.e., the ability to resist squeeze-out is then very well fulfilled by ionic liquids.

The characteristic features of the frictional response across ionic liquids are qualitatively similar to that observed with apolar liquids. A stick-slip behavior is obtained under certain conditions of velocity and load, which is typical of solid-like friction (Figure 2b). The mechanism of molecular reorganisation during the 'slip' phases across apolar liquids has been the matter of intense discussion for many years: the principal distinction being between successive freezing-melting transitions in the nanofilm, or inter-layer (or wall) slip [START_REF] Israelachvili | Shear Properties and Structure of Simple Liquids in Molecularly Thin Films: The Transition from Bulk (Continuum) to Molecular Behavior with Decreasing Film Thickness[END_REF][START_REF] Thompson | Origin of Stick-Slip Motion in Boundary Lubrication[END_REF][START_REF] Klein | Confinement-Induced Phase Transitions in Simple Liquids[END_REF][START_REF] Klein | Simple liquids confined to molecularly thin layers. I. Confinement-induced liquid-to-solid phase transitions[END_REF][START_REF] Alba-Simionesco | Effects of confinement on freezing and melting[END_REF][START_REF] Klein | Frictional Dissipation in Stick-Slip Sliding[END_REF][START_REF] Docherty | Direct evidence for fluid-solid transition of nanoconfined fluids[END_REF][START_REF] Lei | Stick-Slip Friction and Energy Dissipation in Boundary Lubrication[END_REF][START_REF] Rosenhek-Goldian | On the question of whether lubricants fluidize in stick-slip friction[END_REF][START_REF] Jee | Scrutinizing evidence of no dilatancy upon stick-slip of confined fluids[END_REF][START_REF] Israelachvili | On the conformational state of molecules in molecularly thin shearing films[END_REF][START_REF] Rosenhek-Goldian | Israelachvili and Drummond: Lubricant films do not fluidize in intermittent stick-slip friction[END_REF]. It has been argued that, in the case of ionic liquids, slip (or 2D in-plane melt) is likely to be more favourable than full-film (3D) melting during the slip phase [START_REF] Smith | Quantized friction across ionic liquid thin films[END_REF][START_REF] Smith | Molecular Friction Mechanisms Across Nanofilms of a Bilayer-Forming Ionic Liquid[END_REF].

Several papers report "quantized friction" when measuring the friction force as a function of the load and of the number of layers in the film [START_REF] Perkin | Layering and shear properties of an ionic liquid, 1-ethyl-3-methylimidazolium ethylsulfate, confined to nano-films between mica surfaces[END_REF][START_REF] Smith | Quantized friction across ionic liquid thin films[END_REF][START_REF] Werzer | Ionic liquid nanotribology: mica-silica interactions in ethylammonium nitrate[END_REF][START_REF] Elbourne | Adsorbed and near-surface structure of ionic liquids determines nanoscale friction[END_REF]. The quantized description refers to the discrete multi-valued nature of the friction force corresponding to each single value of applied load; this phenomenon is not unique to ionic liquids and will arise in every case of an oscillatory normal force law. As we clearly see in Figure 2c, each liquid composition corresponds to a different linear friction-load relationship. The variation of the intersect can be understood as the adhesion contribution, that increases when the number of layers in the film decreases, and is also a feature shared with apolar molecules [START_REF] Gee | Liquid to solidlike transitions of molecularly thin films under shear[END_REF][START_REF] Kumacheva | Simple liquids confined to molecularly thin layers. II. Shear and frictional behavior of solidified films[END_REF].

There are, however, two major differences between the frictional responses of ionic liquids and apolar liquids. First, with ionic liquids the friction coefficients (i.e., slope in the friction/load law) is also quantized and increases monotonically when the number of layers decreases, while for apolar liquids variation of friction coefficient with number of layers is barely measurable. Secondly, friction coefficients values can be much lower for ionic liquids than for apolar molecules. For example, it goes from ~0.

1 to ~0.8 for [C 4 C 1 Pyrr][NTf 2 ] [35], from ~0.007 to ~0.5 for [C 10 C 1 Pyrr][NTf 2 ] [36]
, while it is ~1.1 for octamethylcyclotetrasiloxan (OMCTS) [START_REF] Perkin | Layering and shear properties of an ionic liquid, 1-ethyl-3-methylimidazolium ethylsulfate, confined to nano-films between mica surfaces[END_REF] (all between mica in SFB). This relatively low resistance to shear (condition (i)) also makes ionic liquids promising lubricants.

Why is the frictional response of ionic liquids different to that of apolar liquids? It is well established that the ordered layers in a confined film of ionic liquid are distinctly different to that of an apolar liquid, with oscillations in the charge density away from the surface corresponding to alternating layers of excess cation-density and excess anion-density and strongly dependent on the surface polarity and charge [START_REF] Fedorov | Ionic Liquids at Electrified Interfaces[END_REF]. Successive layers away from the boundary walls can have differing composition-ratio of cations to anions-in contrast to the uniform composition in a pure apolar liquid.

The influence of this structure on the lubrication properties was illuminated by comparisons made between measurements with dry and wet ionic liquids. Figure 2d,e shows measurements made with

[C 10 C 1 Pyrr][NTf 2 ]
, which is organized in bilayers [START_REF] Smith | Monolayer to Bilayer Structural Transition in Confined Pyrrolidinium-Based Ionic Liquids[END_REF][START_REF] Smith | Molecular Friction Mechanisms Across Nanofilms of a Bilayer-Forming Ionic Liquid[END_REF], in dry or wet conditions. A key finding of this study is that the friction coefficient for a single bilayer is independent of the water content, whereas it is enhanced by more than an order of magnitude for two bilayers when the liquid is wet. If one makes the reasonable assumption that the water molecules preferentially sits close to the charged heads of the cations, we can deduce that the sliding plane is on the alkyl chains for the single bilayer, while it is on the middle ionic region for the two bilayers. Because of the strong binding of the charged groups on the charged surface, and the variation in friction coefficient with film thickness (number of bilayers present), interlayer slip is expected to be more likely than wall slip. Depending on the number of layers separating the solid, new possible sliding planes appear with different compositions (alkyl chains or charged groups), leading to different resistance to sliding.

Even in the boundary lubrication regime, where the solid surfaces are separated by a liquid film of only a few molecular diameters thickness, one should keep in mind that friction processes usually involve macroscopic dimensions in the plane of the contact. Friction thus results from the collective response of the whole film, and this has been stressed by the observation of "microslip" events, i.e., temporary reduction in friction, that can sometimes become avalanches, i.e., collective slip [START_REF] Espinosa-Marzal | Microslips to "Avalanches" in Confined, Molecular Layers of Ionic Liquids[END_REF] (Figure 3b,c). It is notable that such collective phenomena are determined by molecular features and in particular the structure and dynamics of the most highly confined regions of a macroscopic contact. In considering this interesting matter, we note that a similar scenario arises with the (normal) oscillatory structural force. In that case the oft-posed question is: even if the molecules are locally ordered, a geometry involving curved surfaces (e.g., crossed cylinders, or sphere on flat) means that the liquid film thickness increases continuously with distance away from the point of closest approach.

At one location the film thickness may correspond to an integer number of molecular diameters, while at another point this is not the case. So why do we see an oscillating force? As was addressed long ago in the early measurements of such forces, the (perhaps counterintuitive) result of integrating all the forces across the contact returns an oscillatory function with parameters identical to that of plane-parallel approach with a phase shift [START_REF] Horn | Direct measurement of structural forces between two surfaces in a nonpolar liquid[END_REF]. Heuristically, we can understand that the contribution of the thinnest film is most substantial. It is clear that understanding the emergence of collective slip and the precise features of the stick-slip pattern is the key to quantitatively bridge the molecular scale to the observed frictional behavior.

Given the strong influence of molecular structure and orientation on the friction and dissipation across molecular films, it is clear that altering the molecular architecture and chemistry is a route to tune friction. For example, it has been shown that structural and friction forces vary when using ion pairs of different size and flexibility (alkyl chain length) or symmetry (bilayers structure), or different interactions [START_REF] Hayes | At the interface: solvation and designing ionic liquids[END_REF][START_REF] Sweeney | Effect of ion structure on nanoscale friction in protic ionic liquids[END_REF][START_REF] Federici Canova | Shear dynamics of nanoconfined ionic liquids[END_REF][START_REF] Hoth | Force microscopy of layering and friction in an ionic liquid[END_REF][START_REF] De Wijn | Effects of molecule anchoring and dispersion on nanoscopic friction under electrochemical control[END_REF] (see commonly used ions in Figure 1). The range of ionic liquids available commercially has grown substantially over the recent few years; many systems still have to be explored in order to discover the extent to which behavior is universal or, on the contrary, to discover new properties and possibilities.

So far, the role of surface chemistry has been investigated only a little and is far from being fully understood. This is mainly because the criterion for surfaces to possess atomic-scale smoothness precludes the possibility of systematically measuring behaviour at single asperity contacts for a wide range of materials. However, recent studies have demonstrated how ionic liquid structure and dynamics may vary significantly according to the confining surfaces. For example, oscillatory normal force profiles for Au(100) have been found to be less pronounced than for mica, while at comparable surface potentials [START_REF] Kramer | Dynamic shear force microscopy of confined liquids at a gold electrode[END_REF]. Nanoscopic friction response can also depends greatly on surface specific molecular anchoring effects [START_REF] De Wijn | Effects of molecule anchoring and dispersion on nanoscopic friction under electrochemical control[END_REF]. As detailed in Section 4, surface chemistry becomes increasingly important when investigating potential or field dependent effects.

Small Fractions of Molecular Liquid Can Dramatically Influence Lubrication of Ionic Liquids

It has long been known that physical properties (e.g., viscosity, conductivity) of ionic liquids are dramatically altered by the presence of small amounts of water, and that removal all water is challenging. Ideal model experiments can be performed under rather dry conditions (typically 100 ppm water for the less hydrophilic ionic liquids), achieved by drying the liquid under vacuum and elevated temperature prior to experiments which are then performed in a closed chamber containing a desiccant and purged with a dry gas. However the hygroscopic nature of most ionic liquids and the ambient humidity involved in most application environments mean that it becomes important to understand the influence of water and other molecular components in ionic liquids.

Solid surfaces are generally charged, and a small water concentration in the bulk can indeed lead to high interfacial concentrations under confinement [START_REF] Fajardo | Water in Ionic Liquid Lubricants: Friend and Foe[END_REF].

Several studies report systematic measurements by varying the water concentration. Regarding the normal force, it has been shown that the structural force profile is significantly modified. An increase of the layer thicknesses is observed for some liquids, which is intuitively rationalized by the intercalation of water molecules, or "swelling" of the ions (Figure 3d) [START_REF] Smith | Molecular Friction Mechanisms Across Nanofilms of a Bilayer-Forming Ionic Liquid[END_REF][START_REF] Espinosa-Marzal | Microslips to "Avalanches" in Confined, Molecular Layers of Ionic Liquids[END_REF][START_REF] Espinosa-Marzal | Ionic Liquids Confined in Hydrophilic Nanocontacts: Structure and Lubricity in the Presence of Water[END_REF]. More intriguing is the thinning of the layers for other liquids, due to subtle structural changes like molecule reorientations or composition changes [START_REF] Espinosa-Marzal | Microslips to "Avalanches" in Confined, Molecular Layers of Ionic Liquids[END_REF][START_REF] Fajardo | Water in Ionic Liquid Lubricants: Friend and Foe[END_REF][START_REF] Espinosa-Marzal | Ionic Liquids Confined in Hydrophilic Nanocontacts: Structure and Lubricity in the Presence of Water[END_REF]. The extension of the structural force region and the amplitude of adhesive minima can even be enhanced, interpreted as the ability of water to increase surface charge [START_REF] Cheng | Characterizing the Influence of Water on Charging and Layering at Electrified Ionic-Liquid/Solid Interfaces[END_REF]. Several studies reported a reduction of the maxima, meaning that water facilitates the squeeze-out of the ionic liquid (Figure 3a) [START_REF] Espinosa-Marzal | Microslips to "Avalanches" in Confined, Molecular Layers of Ionic Liquids[END_REF][START_REF] Fajardo | Water in Ionic Liquid Lubricants: Friend and Foe[END_REF][START_REF] Espinosa-Marzal | Ionic Liquids Confined in Hydrophilic Nanocontacts: Structure and Lubricity in the Presence of Water[END_REF].

The presence of water also affects the dynamics within the layers [START_REF] Espinosa-Marzal | Microslips to "Avalanches" in Confined, Molecular Layers of Ionic Liquids[END_REF][START_REF] Fajardo | Water in Ionic Liquid Lubricants: Friend and Foe[END_REF][START_REF] Espinosa-Marzal | Ionic Liquids Confined in Hydrophilic Nanocontacts: Structure and Lubricity in the Presence of Water[END_REF]. The squeeze-out transition has been found to be much faster with water (Figure 3d). However water also tends to significantly increase friction; perhaps a counterintuitive result. Experiments with two bilayers of

[C 10 C 1 Pyrr][NTf 2 ]
show an increase of the friction coefficient by more than an order of magnitude when water is present within the layers [START_REF] Smith | Molecular Friction Mechanisms Across Nanofilms of a Bilayer-Forming Ionic Liquid[END_REF] (Figure 2e). Similarly, friction response of dry [C 6 C 1 Im][FAP] has been found below instrumental sensitivity, whereas friction is substantial in wet conditions [START_REF] Espinosa-Marzal | Microslips to "Avalanches" in Confined, Molecular Layers of Ionic Liquids[END_REF] (Figure 3b,c). This trend has been confirmed by fluorescence anisotropy experiments [START_REF] Thomaz | The Influence of Water on the Alkyl Region Structure in Variable Chain Length Imidazolium-Based Ionic Liquid/Water Mixtures[END_REF] and by simulations [START_REF] Fajardo | Water in Ionic Liquid Lubricants: Friend and Foe[END_REF]. Humidity can even change the shape of the friction response: adding water in

[C 2 C 1 Pyrr][NTf 2
] can induce a transition from stick-slip behavior to continuous sliding (Figure 3b,c).

Such substantial effects of water on the lubricating performance of ionic liquids may at first sight appear problematic for applications, as moisture is present almost everywhere and lubricated regions usually cannot be perfectly sealed. However it is important to note that the majority of the above molecular-resolution studies were performed with highly hydrophobic, i.e., water-immiscible (though nonetheless hygroscopic) ionic liquids-motivated by the need to approach the water-free limit under laboratory conditions-and therefore the presence of water in these fluids is likely to be more substantial than for more hydrophilic (water-miscible) ionic liquids. It will be important to compare in future the effect of water on hydrophilic vs. hydrophobic ionic liquids. Relatedly, ions containing corrosive and toxic halogens are problematic for applications, and there is a real need for "green" lubricants [START_REF] Shah | Boron in Tribology: From Borates to Ionic Liquids[END_REF][START_REF] Dong | The synthesis and tribological properties of dicarboxylic acid ionic liquids[END_REF]. Designing ionic liquids which are at the same robust to the inclusion of small amounts of water from the environment, environmentally friendly and with desired lubrication ability is a present and important challenge.

Another strategy to tackle this difficulty is to use ionic liquids as additives in solutions, such that the mixture is not sensitive to ambient humidity. This approach is also motivated by the high current cost of ionic liquids compared to usual lubricants and oils. The main challenge is then: could it be possible to achieve in solution lubricating performances which are comparable to those in pure state?

In this respect, measurements of the structural force for mixtures of [C 4 C 1 Pyrr][NTf 2 ] and propylene carbonate seem promising [START_REF] Smith | Switching the Structural Force in Ionic Liquid-Solvent Mixtures by Varying Composition[END_REF]. While the concentration of ionic liquid has been varied continuously, the thickness of the layers abruptly change from the pure propylene carbonate behavior to the pure ionic liquid behavior at a threshold mole fraction of ~40% (Figure 3e). There is, therefore, a wide range of concentrations for which the pure ionic liquid structure is preserved, and one can expect a similar behavior for the frictional response. Propylene carbonate is not a realistic molecular solvent for lubricant applications because of its volatility, instead ionic liquid/oil mixtures would be more suitable candidate mixtures. It is important to note, also, that the polarity and permittivity of the polar vs. nonpolar solvents is likely to impact the structures formed. Studies in this direction showed that, if the compounds are well chosen, the ions can maintain a boundary layer and so preserve good lubricant properties; this is likely to be aided by the surface activity of the ions which leads to a higher concentration at the interface than in the bulk [START_REF] Sweeney | Near surface properties of mixtures of propylammonium nitrate with n-alkanols 2. Nanotribology and fluid dynamics[END_REF][START_REF] Li | Addition of low concentrations of an ionic liquid to a base oil reduces friction over multiple length scales: a combined nano-and macrotribology investigation[END_REF]. One could ask: what is the advantage of ionic liquid additives, compared to the already-used surfactant additives? The main difference is that in their pure form under ambient conditions ionic liquids are liquid, by definition, while surfactants are generally solid. Surfactants form aggregates, or even (liquid) crystal phases, at sufficiently high concentration, therefore as solvent fraction decreases (typically by evaporation or oxidation of solvent) the surfactant content can increase, eventually to a problematic level. Such problems are avoided for ionic liquids, which perform effectively at high concentrations. Dealing with mixtures remains challenging in general and unexpected behaviors can emerge [START_REF] Smith | Influence of Lithium Solutes on Double-Layer Structure of Ionic Liquids[END_REF][START_REF] Hjalmarsson | Effect of Lithium Ions on Rheology and Interfacial Forces in Ethylammonium Nitrate and Ethanolammonium Nitrate[END_REF]; a wide and careful survey of mixtures of ionic liquids with low-volatility molecular solvents and the resulting interfacial properties will be essential for progress in this direction.

Ionic Liquids Can Be Manipulated to Externally Switch Friction

The previous sections focused on "passive" lubrication, i.e., systems for which the lubrication properties are determined by the choice of chemical compounds (lubricant, surface). There is also a strong interest for "smart" lubricants, whose properties could be switched on demand. An external and reversible control of friction would be very useful for applications like micro/nanoelectromechanical systems (MEMS/NEMS), where the production or the stopping of a motion could be performed more efficiently by changing the friction coefficient in addition or instead of the load. Ionic liquids are very promising systems for this purpose, because charges in general can be manipulated by external electric stimuli (electroactive lubrication), and ionic liquid in particular have wide electro-chemical windows.

Achieving such external control is also an opportunity to build a better understanding of the molecular mechanisms of friction.

Important efforts have been made to interact with the ionic liquid by changing the surface charge.

Numerical simulations showed that modifying the surface charge can affect the liquid composition and its structure, and also the frictional response [START_REF] Fajardo | Electrotunable Lubricity with Ionic Liquid Nanoscale Films[END_REF][START_REF] Fajardo | Electrotunable Friction with Ionic Liquid Lubricants: How Important Is the Molecular Structure of the Ions?[END_REF][START_REF] Capozza | Electrical charging effects on the sliding friction of a model nano-confined ionic liquid[END_REF]. Figure 4b,c shows that the friction-load relationship depends on the surface charge, but that this evolution is not monotonic. At low surface charges, most of the sliding occurs at the liquid/solid interface, and dissipation increases while the first layer of counter-ions on each electrode is more on more coupled to the wall. At high surface charges, this first layer is so rigidly coupled to the solid that inter-layer slip becomes dominant, and this can ultimately lead to superlubricity. Usually, boundary lubrication experiments are performed with dielectric surfaces (mica, silica, sapphire etc.), for which the surface charge is not controlled. It depends on the material, the pH of the solution, and complex specific interactions between at the solid/liquid interface. For instance, mica at no applied potential has a significant negative surface charge due to desorption of surface K + ions into solution. Surface charge control is generally reached by using one or two conductive surfaces (gold, graphite, etc.). Doing so, the control parameter is in fact not the surface charge, but the surface potential. Many experiments in this field have been performed by Colloidal Probe Atomic Force Microscope, because of the difficulty to produce conductive electrodes that are clean and smooth at the molecular scales at the scale of the probe. It has been shown that controlling the surface potential is a way to change the boundary film composition, as revealed by the change of the structural force profile [START_REF] Hayes | Double Layer Structure of Ionic Liquids at the Au(111) Electrode Interface: An Atomic Force Microscopy Investigation[END_REF]. Figure 4a shows friction measurements for [C 4 C 1 Pyrr][FAP] confined between silica and a gold electrode of controlled potential [START_REF] Sweeney | Control of Nanoscale Friction on Gold in an Ionic Liquid by a Potential-Dependent Ionic Lubricant Layer[END_REF]. When a negative potential is applied to the gold electrode, the liquid film is enriched in cations, which produces a lower friction coefficient, presumed to be due to the relatively good lubricating alkyl chains of the cation. The sign of the effect depends on the relative lubricity of the two ions [START_REF] Li | Ionic liquid lubrication: influence of ion structure, surface potential and sliding velocity[END_REF], and the effect is not necessarily monotonic, because of specific interactions or subtle reorientations mechanisms. For example, HOPG surfaces have strong solvophobic interactions with non-polar chains of substrate molecules [START_REF] Li | Combined friction force microscopy and quantum chemical investigation of the tribotronic response at the propylammonium nitrate-graphite interface[END_REF], greatly changing friction response to surface potential change when compared with Gold surfaces [START_REF] Li | An ionic liquid lubricant enables superlubricity to be "switched on" in situ using an electrical potential[END_REF][START_REF] Li | Boundary layer friction of solvate ionic liquids as a function of potential[END_REF]. For a particular choice of the liquid and surface chemistry, it has even been shown that it is possible to switch superlubricity [START_REF] Li | An ionic liquid lubricant enables superlubricity to be "switched on" in situ using an electrical potential[END_REF]. In general, these studies at controlled surface charge are powerful tools to try to link the liquid structure to the frictional behavior and to identify the dissipation processes at play. Another strategy, not explored yet to our knowledge, would be to apply an electric field across the nano-film. Such control can be achieved with a Surface Force Apparatus, for which the metallic mirrors at the back of the mica can be used as electrodes. This method has been used for polyelectrolyte coatings immersed in water and for liquid crystals [START_REF] Drummond | Electric-Flield-Ilnduced Flriction Rleduction and Clontrol[END_REF][START_REF] Kimura | Control of friction coefficient by applying electric fields across liquid crystal boundary films[END_REF][START_REF] Nakano | Effect of confinement on electric field induced orientation of a nematic liquid crystal[END_REF]. In Figure 4d is reproduced the friction-load characteristics between to mica surfaces bearing polyelectrolyte brushes. When an alternating electric field is applied, friction is dramatically reduced because interpenetration of the polymer chains is prevented. Similarly, one can expect for ionic liquids that the electric field induces structural changes, and so modifies the lubrication properties of the film.

Conclusions

In this review, we gather the main findings about frictional behavior of ionic liquids in the boundary lubrication regime. Ionic liquids are promising lubricants because of their ability to form structured boundary layers which are strongly bound on charged or polar surfaces, promoting a low-dissipating inter-layer slip. The main limitations are their sensitivity to ambient moisture, their possible toxicity and their cost, that could be overcame by developing green ionic liquid additives.

Their strong and switchable response to external electrical stimuli make them systems of choice for smart applications [START_REF] Cooper | Tribotronic control of friction in oil-based lubricants with ionic liquid additives[END_REF]. Interesting properties or even new behaviors could emerge from zwiterrionic ionic liquids [START_REF] Bordes | Room-Temperature Zwitterionic Ionic Liquids[END_REF], which have to our knowledge not yet been investigated.

Figure 1 .

 1 Figure 1. (a) Friction is a multi-scale phenomenon: the macroscopic sliding of usual surfaces is slowed down by dissipation involving many mesoscopic asperities, where the lubricant should act at the molecular scale. (b) Schematic and relevant quantities of a typical measurement of friction on a single contact across an ionic liquid. (c) Examples of cations common in ionic liquids, from top to bottom: 1-alkyl-1-methylpyrrolidinium [C n C 1 Pyrr] + , 1-alkyl-3-methylimidazolium [C n C 1 Im] + , phosphonium [R i P] + . (d) Examples of anions common in ionic liquids, from top to bottom: bis[(trifluoromethane)sulfonyl]imide [NTf 2 ] -(also commonly known as TFSI), ethylsulfate [EtSO 4 ] -, tris(pentafluoroethyl)trifluorophosphate [FAP] -.

Figure 2 .

 2 Figure 2. (a) (b) (c) Forces between mica surfaces across a structured nanofilm of [C 4 C 1 Pyrr][NTf 2 ], measured with a Surface Force Balance: (a) oscillatory profile of the normal force rescaled by the radius of curvature of the single-asperity contact (proportional to the interaction energy between parallel plates of the same material at the separation of closest approach, according to the Derjaguin approximation); (b) stick-slip evolution of the lateral force as a function of the lateral displacement, for three different numbers of layers at same load and velocity, indicated by the black circles in (c); (c) kinetic friction force/normal force relationship, showing quantized friction. Figures adapted from [35] -Published by the PCCP Owner Societies. (d) (e) Similar friction measurements, performed with (d) one or (e) two bilayers of [C 10 C 1 Pyrr][NTf 2 ], under dry or wet conditions as indicated in the legends. Water molecules are used as "tracers" to deduce the sliding planes, indicated in the cartoons. Adapted with permission from [36] -Copyright 2014 American Chemical Society.

  Figure 2. (a) (b) (c) Forces between mica surfaces across a structured nanofilm of [C 4 C 1 Pyrr][NTf 2 ], measured with a Surface Force Balance: (a) oscillatory profile of the normal force rescaled by the radius of curvature of the single-asperity contact (proportional to the interaction energy between parallel plates of the same material at the separation of closest approach, according to the Derjaguin approximation); (b) stick-slip evolution of the lateral force as a function of the lateral displacement, for three different numbers of layers at same load and velocity, indicated by the black circles in (c); (c) kinetic friction force/normal force relationship, showing quantized friction. Figures adapted from [35] -Published by the PCCP Owner Societies. (d) (e) Similar friction measurements, performed with (d) one or (e) two bilayers of [C 10 C 1 Pyrr][NTf 2 ], under dry or wet conditions as indicated in the legends. Water molecules are used as "tracers" to deduce the sliding planes, indicated in the cartoons. Adapted with permission from [36] -Copyright 2014 American Chemical Society.

Figure 3 .

 3 Figure 3. (a)-(d) Forces between mica surfaces across structured nanofilms of [C 2 C 1 Im][FAP] and [C 6 C 1 Im][FAP], measured with a Surface Force Apparatus: (a) profile of the normal force rescaled by the radius of curvature of the single-asperity contact; (b) temporal evolution of the friction force in dry condition; (c) temporal evolution of the friction force in wet condition; (d) squeeze-out dynamics. Reprinted with permission from [41] -Copyright 2014 American Chemical Society. (e) Mean thickness of the structured layers for different mixtures of [C 4 C 1 Pyrr][NTf 2 ] in propylene carbonate. Reprinted with permission from [54] -Copyright 2017 by the American Physical Society.

Figure 4 .

 4 Figure 4. (a) Friction force/normal force relationship for [C 4 C 1 Pyrr][FAP] between silica and a gold electrode at different potentials, measured with a Colloidal Probe Atomic Force Microscope. Reprinted with permission from [62] -Copyright 2012 by the American Physical Society. (b) (c) Dependence of the friction force on the load and the electric charge of the confining surfaces obtained numerically: (b) friction force/normal force relationship for two surface charges; (c) friction force/surface charge relationship for two loads (188 MPa in green and 500 MPa in blue). Reproduced with permission from [59]. (d) Friction force/normal force relationship for polyelectrolyte brushes coating mica in a Surface Force Apparatus, with (red circles) and without (black squares) an applied alternating electric field. Reprinted with permission from [67] -Copyright 2012 by the American Physical Society.

Acknowledgments: S.P. and R.L. are supported The Leverhulme Trust (RPG-2015-328) and the ERC (under Starting Grant No. 676861, LIQUISWITCH). R.L. is supported by the EPA Cephalosporin Junior Research Fellowship and Linacre College (University of Oxford). S.P. is grateful for research leave enabled by the Philip Leverhulme Prize.

Conflicts of Interest:

The authors declare no conflict of interest.


G Andrade-Miranda

V Jaouen

V Bourbonne

F Lucia

D Visvikis

P.-H Conze

PURE VERSUS HYBRID TRANSFORMERS FOR MULTI-MODAL BRAIN TUMOR SEGMENTATION: A COMPARATIVE STUDY

Keywords: Vision Transformers, tumor segmentation, multi-modality, hybrid CNN-Transformers models

Vision Transformers (ViT)-based models are witnessing an exponential growth in the medical imaging community. Among desirable properties, ViTs provide a powerful modeling of long-range pixel relationships, contrary to inherently local convolutional neural networks (CNN). These emerging models can be categorized either as hybrid-based when used in conjunction with CNN layers (CNN-ViT) or purely Transformers-based. In this work, we conduct a comparative quantitative analysis to study the differences between a range of available Transformers-based models using controlled brain tumor segmentation experiments. We also investigate to what extent such models could benefit from modality interaction schemes in a multi-modal setting. Results on the publicly-available BraTS2021 dataset show that hybrid-based pipelines generally tend to outperform simple Transformersbased models. In these experiments, no particular improvement using multi-modal interaction schemes was observed.

INTRODUCTION

Deep segmentation models derived from U-Net [START_REF] Ronneberger | U-Net: Convolutional networks for biomedical image segmentation[END_REF] have significantly transformed the field of medical image segmentation due to their ability to learn complex local representations at multiple spatial scales in a data-driven fashion [START_REF] Isensee | nnU-Net: a selfconfiguring method for deep learning-based biomedical image segmentation[END_REF][START_REF] Conze | Abdominal multi-organ segmentation with cascaded convolutional and adversarial deep networks[END_REF][START_REF] Iantsen | Squeeze-and-excitation normalization for brain tumor segmentation[END_REF]. After having established new benchmark performances on image classification tasks, Vision Transformers (ViT) models [START_REF] Dosovitskiy | An image is worth 16x16 words: Transformers for image recognition at scale[END_REF] have recently emerged as the most popular option to replace or complement convolutional neural networks (CNN) for computer vision applications [START_REF] Carion | End-to-end object detection with transformers[END_REF]. Their popularity are now also rapidly growing in medical image analysis [START_REF] Jun | Medical Transformer: Universal brain encoder for 3D MRI analysis[END_REF], especially for medical image segmentation with an exponential growth of related publications in the last year [START_REF] Shamshad | Transformers in medical imaging: A survey[END_REF]. Depending on the type of encoder used in these models, two categories can be identified: pure Transformers-based made of ViT layers only and hybrid-based models composed of both CNN and ViT layers. The first category exploits the global context modeling capability of Transformers to effectively encode the relationships between spatially distant voxels. However, as anatomical structures can substantially vary in scale, they cannot be properly modelled using a set of fixed sub-regions of the image [START_REF] Hatamizadeh | UNETR: Transformers for 3D medical image segmentation[END_REF]. For this reason, hybrid architectures combining the global context modeling ability of Transformers with the CNN inductive bias are also popular [START_REF] Chen | TransUNet: Transformers make strong encoders for medical image segmentation[END_REF][START_REF] Wang | TransBTS: Multimodal brain tumor segmentation using Transformer[END_REF]. CNN layers are used as multi-scale feature extractors, while Transformers capture long-term dependencies among features that would be potentially lost with purely convolutional models. Recently, hierarchical ViT such as Swin Transformers [START_REF] Liu | Swin Transformer: Hierarchical vision Transformer using shifted windows[END_REF][START_REF] Hatamizadeh | Swin UN-ETR: Swin Transformers for semantic segmentation of brain tumors in MR images[END_REF] have also been introduced to overcome these challenges by extracting features at different resolutions while saving the linear computational complexity with respect to image size.

The advantages of pure Transformer-based models over hybrid approaches in medical imaging are not yet clear as of today. Moreover, the exploitation of cross-modality correlations in these models did not receive much attention, leaving aside potentially meaningful information for segmentation purposes. The goal of this paper is therefore to provide insights into the performances of various hybrid and Transformers-based networks in the context of multi-modal tumor segmentation with BraTS, a popular segmentation challenge. The rest of the paper is organized as follows. Sect.2 introduces both ViT and CNN-ViT architectures. Sect.3 presents implementation details as well as the evaluation strategy. Sect.4 explains and discusses the results we obtained. Conclusions and perspectives are given in Sect.5.

METHODS

CNN backbone

The hybrid models we consider in this work are made of ResNet alike encoders at the shallower levels (Fig. 1a) to capture compact features at multiple scales, while deeper levels are encoded with Transformer blocks to model longrange dependencies in a more global manner. Given an input X ∈ R C×H×W ×D with spatial resolution H × W , D as depth dimension (# of slices) and C channels (# of modalities), down-sampling blocks gradually encode input images into a low-resolution/high-level feature representation 8 which is 1 8 of input dimensions and where F represents the number of encoded feature maps [START_REF] Wang | TransBTS: Multimodal brain tumor segmentation using Transformer[END_REF]. An extra max-pooling operation is applied to further reduce the burden of computational complexity as input of the Transformer blocks. Such backbone can be extended to a multi-encoder-based framework, where independent encoders learn intra-modality feature representations meanwhile the Transformer block carries out the inter-modality fusion operation.

F ∈ R F × H 8 × W 8 × D

Transformer block

Since Transformers operate on 1D sequences, we reshape the input X ∈ R C×H×W ×D into a sequence of flattened uniform non-overlapping patches

x v ∈ R N ×C×P 3
, where C can either represent the number of modalities or features maps F , (P, P, P ) is the size of each patch and N = HW D/P 3 is the resulting number of patches, which is also the effective input length of the Transformer. Then, we distinguish three types of Transformer blocks: vanilla ViT, single-stream ViT (Fig. 2a) and multiple-stream ViT (Fig. 2b).

The vanilla ViT block (ViT v ) follows the design proposed in the original ViT paper [START_REF] Dosovitskiy | An image is worth 16x16 words: Transformers for image recognition at scale[END_REF]. First, a linear layer is used to project the flattened patches into a K-dimensional embedding space. Then, a 1D learnable patch position embedding E pos ∈ R N ×k is added to the projected embeddings to retain positional information. L Transformer blocks are then further stacked, comprising h multi-head attention and multi-layer perceptron (MLP) sub-layers. On the other hand, singlestream ViT (ViT s ) layers collectively operate on a concatenation of images modalities [START_REF] Devlin | BERT: Pre-training of deep bidirectional Transformers for language understanding[END_REF] by adding an extra modal-type embedding to the learnable token-position and projected embeddings. This design enables early and unconstrained fusion of cross-modal information. In multiple-stream ViT (ViT m ) [START_REF] Lu | VilBERT: Pretraining task-agnostic visiolinguistic representations for vision-and-language tasks[END_REF], modalities are first handled by independent Transformer blocks. The resulting representations are then fed to a co-

A A 1 A 2 A 3 B 1 B 2 B 3 C 1 C 2 C 3 D 1 D 2 D 3
Linear attention Transformer layer where intra-modal interactions are alternated with inter-modal interactions.

Multi-level feature aggregation decoder

Decoding towards the segmentation mask space is handled using CNN up-sampling with multi-level feature aggregation.

The building decoding block is shown in Fig. 1b. For hybrid models, we first project back the output of the Transformer block in the layer

L, z L ∈ R N ×k to k × H 16 × W 16 × D 16 .
Then, we apply a transpose 3D convolution to go back to the original output of the CNN encoder k × H 8 × W 8 × D 8 . The upsampling blocks reduce the channel dimension of the decoder to match with the dimension of the skip-connections coming from the encoding part, decreasing computational complexity at the same time. Lastly, we perform a progressive feature up-sampling to gradually recover the full pixel-level segmentation segmentation Y ∈ R H×W ×D . On the other hand, the ViT-based architecture follows the same decoder proposed in [START_REF] Hatamizadeh | UNETR: Transformers for 3D medical image segmentation[END_REF] that only differs from the aforementioned one in the operation needed to reshape the features from the multiple resolutions of the Transformers-based encoder.

Hybrid and Transformers-based architectures

The networks presented in this work follow a U-shaped design in which the extracted feature representations from encoder layers are fused with their decoder counterparts by concatenation for finer segmentation masks with richer spatial details through skip-connections. The proposed models are denoted as CNN+ViT τ -B/P, MCNN+ViT τ -B/P and ViT τ -B/P, where the subscript τ represents the type of ViT block, B indicates the base ViT model configuration, P the patch size and M stands for multi-encoder based CNN. From the different possible configurations, we derive a total of seven mod-

C × H × W × D F 1 × H × W × D F 2 × H 2 × W 2 × D 2 F 3 × H 4 × W 4 × D 4 F 4 × H 8 × W 8 × D 8 H × W × D F 2 × H × W × D F 3 × H 2 × W 2 × D 2 F 4 × H 4 × W 4 × D 4 4F 4 × H 4 × W 4 × D 4 4F 3 × H 2 × W 2 × D 2 4F 2 × H × W × D k × H 8 × W 8 × D 8 F 4 × H 16 × W 16 × D 16 k × H 8 × W 8 × D 8 4 × (C × H × W × D) 4 × (F 1 × H × W × D) 4 × F 2 × H 2 × W 2 × D 2 4 × F 3 × H 4 × W 4 × D 4 H × W × D 4 × F 4 × H 8 × W 8 × D 8 4 × F 4 × H 16 × W 16 × D 16 VIT VIT VIT 4F 4 × H 4 × W 4 × D 4 4F 3 × H 2 × W 2 × D 2 4F 2 × H × W × D H × W × D C × H × W × D v

Implementation details

We implemented our models using PyTorch and MONAI 1 . Hybrid and Transformers-based models were trained using Nvidia A6000 and Titan RTX GPUs. All models were trained for a total of 150 epochs, with a batch size of 2 and NovoGrad For a fair comparison, we followed both pre-processing and ET WT TC Avg. Fold1 0.893 0.918 0.900 0.904 0.886 0.925 0.912 0.908 0.889 0.924 0.906 0.907 0.895 0.920 0.920 0.911 Fold2 0.875 0.916 0.913 0.901 0.893 0.930 0.912 0.912 0.882 0.905 0.911 0.899 0.876 0.914 0.920 0.903 Fold3 0.879 0.909 0.919 0.902 0.878 0.929 0.924 0.910 0.890 0.928 0.931 0.916 0.889 0.927 0.937 0.918 Fold4 0.901 0.919 0.923 0.914 0.906 0.920 0.934 0.920 0.905 0.922 0.940 0.922 0.904 0.918 0.923 0.915 Fold5 0.853 0.920 0.893 0.889 0.877 0.928 0.915 0.907 0.870 0.916 0.917 0.901 0.867 0.920 0.909 0.898 Avg. 0.880 0.916 0.909 0.902 0.888 0.926 0.919 0.911 0.887 0.919 0.921 0.909 0.886 0.920 0.922 0.909

Transformers-based models (ViT)

ViT v -B/16
ViT s -B/16 VIT m -B/16 UNETR [9] Fold1 0.854 0.893 0.833 0.860 0.836 0.867 0.778 0.827 0.843 0.851 0.781 0.825 0.862 0.889 0.867 0.873 Fold2 0.840 0.902 0.827 0.856 0.838 0.850 0.811 0.833 0.841 0.870 0.815 0.842 0.857 0.896 0.862 0.872 Fold3 0.833 0.889 0.824 0.848 0.814 0.849 0.810 0.824 0.839 0.846 0.815 0.833 0.847 0.887 0.862 0.865 Fold4 0.874 0.898 0.839 0.870 0.852 0.844 0.810 0.835 0.839 0.852 0.782 0.824 0.875 0.903 0.889 0.889 Fold5 0.836 0.901 0.859 0.866 0.804 0.866 0.791 0.820 0.805 0.860 0.792 0.819 0.840 0.905 0.854 0.866 Avg. 0.848 0.896 0.836 0.860 0.829 0.855 0.800 0.828 0.833 0.856 0.797 0.829 0.856 0.896 0.867 0.873 Table 2. Five-fold cross-validation benchmark in terms of averaged Dice score for hybrid and Transformers-based models. ET, WT and TC respectively denote enhancing tumor, whole tumor and tumor core. Best results are in bold. data augmentation strategies employed in nnU-Net [START_REF] Isensee | nnU-Net: a selfconfiguring method for deep learning-based biomedical image segmentation[END_REF] for all the implemented models.

Evaluation of predicted segmentation

We first splited the BraTS2021 training phase data with a 95:5 ratio to get both training and test sets. Over the new training set, we used 5-fold cross-validation with fixed 80:20 split for all experiments and evaluated the performance of our models using averaged Dice (DSC) scores. The evaluation was carried out using eight different models: four hybrids-based, three Transformer-based and UNETR [START_REF] Hatamizadeh | UNETR: Transformers for 3D medical image segmentation[END_REF] as baseline. The results of the cross-validation are summarized in Tab.2 meanwhile Fig. 4 illustrates qualitative segmentation results for the best hybrid and Transformer-based models, over the test set.

RESULTS AND DISCUSSION

Results provided in Tab.2 show that MCNN+ViT v -B/1 achieves the highest overall Dice score (0.911). However, its MCNN+ViT s -B/1 and MCNN+ViT m -B/1 variants are not far (0.909 for both). Concerning the tumor sub-regions, MCNN+ViT v -B/1 is only outperformed by its multi-modal variants for TC by 0.002 and 0.003 respectively, which indicates no particular improvements from the use of modality interaction schemes. For all folds, multi-encoder based frameworks outperformed the single encoder approach by around 0.009. Another interesting finding is the fact that all the hybrids models outperformed the Transformers-based ones by approximately 0.038. This suggests that ViT blocks learn more helpful cross-modal representation when they rely on previous image feature extraction through a CNN backbone. The highest overall Dice score in the Transformers-based model was obtained with UNETR (0.873), closely followed by ViT v -B/16 (0.860). Both models have the same encoder but they slightly differ in the decoding process as well as for skip-connections. This difference may be due to the design of the decoder, for which further studies out of the scope of the present work should be performed. Surprisingly, ViT s -B/16 and ViT m -B/16 are the worst ranked models with Dice scores of 0.828 and 0.829. This could be linked to the lack of multiscale information, making it hard for the modality interaction schemes to discover meaningful cross-modal representations.

In these experiments, the hybrid MCNN+ViT v -B/1 model achieved the best result. Learning modality-specific encoding using multi-encoder-based frameworks improved delineation performance (Fig. 4) but at the expense of a higher computational cost, while pure ViT s -B/16 and ViT m -B/16 performed worse, suggesting that current modality interaction strategies may in fact be detrimental to model robustness when implemented in a pure Transformers-based fashion.

CONCLUSION

In this work, we compared a variety of ViT and hybrid ViT-CNN architectures in the context of multi-modal tumor segmentation using the BraTS dataset. The major conclusion of this preliminary work is that it seems better to take advantage of hybrid methods exploiting the local inductive bias from CNN encoders at shallower levels but also selfattention mechanisms able to capture longer-range dependencies among features, rather than using pure Transformersbased architectures. While single and multiple cross-modal interaction models did not achieve the highest performances in these experiments, future works will help us to better understand this issue, with the objective of achieving parameterefficient architectures in a multi-modal context. The presented results are preliminary, and more experiments are indeed needed to generalize our conclusions to other datasets. Investigating pre-trained models, robustness to varying data regimes (from low-to high-) and hierarchical Transformers should also deserve more in-depth investigations.

Fig. 1 .

 1 Fig. 1. Basic building blocks used to build the CNN backbone encoder: (a) down-and (b) up-sampling ResNet blocks.

Fig. 2 .

 2 Fig. 2. Two possible modality interaction schemes: (a) single-stream (ViT s ), (b) multiple-stream (ViT m ) interactions.

Fig. 3 . 9 ] 1 .

 391 Fig. 3. Hybrid CNN-Transformers models: (left) CNN+ViT v -B/P, (middle) MCNN+ViT τ -B/P, (right) ViT τ -B/P.

  The magnetic resonance (MR) images used for this work are from the BraTS2021 training dataset[START_REF] Baid | The RSNA-ASNR-MICCAI BraTS 2021 benchmark on brain tumor segmentation and radiogenomic classification[END_REF][START_REF] Bjoern H Menze | The multimodal brain tumor image segmentation benchmark (BraTS)[END_REF][START_REF] Bakas | Advancing the cancer genome atlas glioma MRI collections with expert segmentation labels and radiomic features[END_REF]. The related BraTS challenge focuses on the evaluation of state-of-the-art methods for the segmentation of intrinsically heterogeneous brain glioblastomas. The training set contains 1251 subjects. The 3D MR scans include four modalities: native (T1), postcontrast T1-weighted (T1Gd), T2-weighted (T2) and T2 fluid attenuated inversion recovery (T2-FLAIR). Annotations include Gadolinium-enhancing tumor region (ET), peritumoral edematous/invaded tissues and the necrotic tumor core that are combined to obtain three sub-regions: tumor core (TC), whole tumor (WT) and enhanced tumor (ET). All 3D volumes are skull-striped and resampled to 1mm 3 isotropic resolution with an input image size of 240 × 240 × 155 voxels.

1Fig. 4 .

 4 Fig. 4. Visual segmentation results: (left) ground truth, (middle) UNETR [9] and (right) MCNN+ViT v -B/P. The green, red and blue colors respectively correspond to WT, TC and ET glioblastoma sub-regions.

This work benefited from state aid managed by the National Research Agency under the Future Investment Program bearing the reference ANR-17-RHUS-0005 (FollowKnee project). This work was also partially funded by France Life Imaging (grant ANR-11-INBS-0006).