A Review of Gas-Surface Interaction Models for Orbital Aerodynamics Applications
Sabrina Livadiotti, Nicholas H. Crisp, Peter C.E. Roberts, Stephen D. Worrall, Vitor T.A. Oiko, Steve Edmondson, Sarah J. Haigh, Claire Huyton, Katharine L. Smith, Luciana A. Sinpetru, Brandon E. A. Holmes, Jonathan Becedas, Rosa María Domínguez, Valentín Cañas, Simon Christensen, Anders Mølgaard, Jens Nielsen, Morten Bisgaard, Yung-An Chan, Georg H. Herdrich, Francesco Romano, Stefanos Fasoulas, Constantin Traub, Daniel Garcia-Almiñana, Silvia Rodriguez-Donaire, Miquel Sureda, Dhiren Kataria, Badia Belkouchi, Alexis Conte, Jose Santiago Perez, Rachel Villain, Ron Outlaw
AA Review of Gas-Surface Interaction Models for OrbitalAerodynamics Applications
Sabrina Livadiotti* a , Nicholas H. Crisp a , Peter C.E. Roberts a , Stephen D. Worrall a , Vitor T.A.Oiko a , Steve Edmondson a , Sarah J. Haigh a , Claire Huyton a , Katharine L. Smith a , Luciana A.Sinpetru a , Brandon E. A. Holmes a , Jonathan Becedas b , Rosa Mar´ıa Dom´ınguez b , Valent´ınCa˜nas b , Simon Christensen c , Anders Mølgaard c , Jens Nielsen c , Morten Bisgaard c , Yung-AnChan d , Georg H. Herdrich d , Francesco Romano d , Stefanos Fasoulas d , Constantin Traub d ,Daniel Garcia-Almi˜nana e , Silvia Rodriguez-Donaire e , Miquel Sureda e , Dhiren Kataria f , BadiaBelkouchi g , Alexis Conte g , Jose Santiago Perez g , Rachel Villain g and Ron Outlaw h a The University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom b Elecnor Deimos Satellite Systems, Calle Francia 9, 13500 Puertollano, Spain c GomSpace AS, Langagervej 6, 9220 Aalborg East, Denmark d University of Stuttgart, Pfa ff enwaldring 29, 70569 Stuttgart, Germany e UPC-BarcelonaTECH, Carrer de Colom 11, 08222 Terrassa, Barcelona, Spain f Mullard Space Science Laboratory (UCL), Holmbury St. Mary, Dorking, RH5 6NT, United Kingdom g Euroconsult, 86 Boulevard de S´ebastopol, 75003 Paris, France h Christopher Newport University Engineering, Newport News, Virginia 23606, United States
Abstract
Renewed interest in Very Low Earth Orbits (VLEO) - i.e. altitudes below 450 km - has led toan increased demand for accurate environment characterisation and aerodynamic force predic-tion. While the former requires knowledge of the mechanisms that drive density variations inthe thermosphere, the latter also depends on the interactions between the gas-particles in theresidual atmosphere and the surfaces exposed to the flow. The determination of the aerodynamiccoe ffi cients is hindered by the numerous uncertainties that characterise the physical processes oc-curring at the exposed surfaces. Several models have been produced over the last 60 years withthe intent of combining accuracy with relatively simple implementations. In this paper the mostpopular models have been selected and reviewed using as discriminating factors relevance withregards to orbital aerodynamics applications and theoretical agreement with gas-beam experi-mental data. More sophisticated models were neglected, since their increased accuracy is gener-ally accompanied by a substantial increase in computation times which is likely to be unsuitablefor most space engineering applications. For the sake of clarity, a distinction was introducedbetween physical and scattering kernel theory based gas-surface interaction models. The phys-ical model category comprises the Hard Cube model, the Soft Cube model and the Washboardmodel, while the scattering kernel family consists of the Maxwell model, the Nocilla-Hurlbut-Sherman model and the Cercignani-Lampis-Lord model. Limits and assets of each model havebeen discussed with regards to the context of this paper. Wherever possible, comments havebeen provided to help the reader to identify possible future challenges for gas-surface interactionscience with regards to orbital aerodynamic applications. Keywords:
Gas-Surface Interaction, Very Low Earth Orbit, Orbital Aerodynamics
Preprint submitted to Progress in Aerospace Sciences November 24, 2020 a r X i v : . [ phy s i c s . s p ace - ph ] N ov i ffi culties in modelling the interaction of the near-Earth aerodynamic environment withsatellites in Low Earth Orbit (LEO) are due to a lack of knowledge on the mechanisms that de-termine the thermosphere total density variation, the magnitude and the direction of the thermo-spheric wind vector and the dynamics of the Gas-Surface Interaction (GSI). These uncertaintiesa ff ect the computation of the acceleration that drag - the main source of perturbation for altitudesbelow 600 km [1] - exerts on satellites: a D = − ρ V rel C D S re f m V rel | V rel | (1)where S re f is the reference surface adopted to perform the computation and m is the satellite’smass, often the only parameter known with substantial accuracy unless any propellant consump-tion needs to be acknowledged. In Eq. 1 uncertainties are found in the assessment of the totaldensity ( ρ ), the aerodynamic drag coe ffi cient ( C D ) and the satellite velocity with regards to therotating atmosphere (cid:0) V rel (cid:1) . Since these sources of uncertainties are mutually linked, any attemptto discuss them separately is improper. However, the complexity of the problem demands someform of simplification. Therefore, challenges encountered in estimating ρ and V rel , whose vari-ations are generally associated with fluctuations in the thermosphere environment, will not betreated in this review. Consequently, the key factors that for a given velocity of the flow con-tribute to determination of the dynamic pressure q (Eq. 2) will be disregarded to devote moreattention to those engineering variables that can be modified through proper design and materialsselection: q ( t ) = ρ ( t ) V rel ( t ) (2)For the reader’s knowledge, comprehensive works covering the mechanisms a ff ecting the esti-mate of dynamic pressure can be found in [2–8].In the following sections of this paper, the e ff ect of GSI dynamics on drag evaluation andcomputation of the aerodynamic coe ffi cients will be discussed. Some information regarding theaerodynamic regime experienced by satellites in LEO - and especially in Very Low Earth Orbit(VLEO) - will be provided in an attempt to create an adequate background for the discussion.Attention will be focused on statistical and physical GSI models which have obtained consider-able success both in engineering applications and surface science for their capability to describethe complex processes occurring at the surface with relative simplicity. In this regard, the inten-tion is to create continuity with a recent review paper on the topic by Mostaza Prieto et al. [9],which focuses on classical analytical models used for aerodynamic computation in LEO. Gas-beam experimental results conducted in the physical regimes of interest for this paper will finallybe presented, and the behaviour of the models described will be discussed, wherever possible,against the identified trends in scattering. The objective is thus to highlight the points of strengthand the limits of the theoretical models against the available experimental data. In this way pos-sible feature developments can be discussed and hopefully a reasonable picture of the di ffi cultiesencountered in approaching the GSI problem for orbital aerodynamics can be provided.Increased knowledge of the interaction mechanisms occurring in the gas-solid phase systemis crucial not only for scientific achievement, but also for the possibility of improving aerody-namic performance of spacecraft operating in VLEO [10, 11]. This would reflect in increasedconfidence in assessing the advantages and the drawbacks of employing aerodynamic torquesfor orbit [12–18] and attitude control purposes [19–28]. Overestimating or underestimating theaerodynamic torques induced by the actuation of aerodynamic control surfaces has an impact on2he altitude range for which aerodynamic manoeuvring is expected to be feasible. This seems rel-evant especially for missions operating in periods of low solar activity, when the thermosphericdensity values at altitudes above 200 km are significantly lower than those expected during highsolar activity [9]. In terms of attitude control implementation, the achievable aerodynamic con-trol authority about the roll, pitch and yaw body axes drives the design of the controller selected.This is especially true if conventional actuators (e.g. reaction wheels, magnetorquers) are meantto be used in synergy with the designated control surfaces. Undesired aerodynamic torques maycounteract the control action of the wheels, disturbing the attitude task and eventually leading tosaturation. Similarly, aerodynamic orbit control [29], formation flying [30–33] and rendezvousmanoeuvres [34, 35] would significantly benefit from any improvement in GSI models, espe-cially with regards to the possibility of producing control torques in the direction perpendicularto the orbit plane. A reliable estimation of the aerodynamic coe ffi cients is also fundamental inthe evaluation of the impact that aerodynamic based manoeuvres may have on the rate of decayof satellites in orbit. This knowledge could potentially be useful for a better prediction of thesatellite re-entry trajectory [16, 18] or to achieve a better knowledge of the expected aerody-namic forces and torques induced on the ram surfaces during controlled re-entry in atmosphere.This knowledge represents a considerable advantage even for spacecraft that are already in orbit,assuming that the materials employed for the external surfaces are known and that a good pre-diction of the environmental conditions is achiavable. Nevertheless, a better knowledge of thephenomena occuring at the surface could potentially drive a more rational design of the satellitegeometry according to the desired aerodynamic performances. The same geometry is indeedexpected to provide a di ff erent aerodynamic behaviour with varying scattering re-emission pat-terns. Comparably, the design of Atmospheric Breathing Electric Propulsion (ABEP) systemsis driven by the aerodynamic performance expected for the materials employed [36]. Any im-provements in the reliability of the GSI models may translate in a more confident definition ofoptimal performance ranges and it can possibly pave the way for new design criteria.
1. The Aerodynamic Environment
At the altitudes where VLEO satellites orbit, i.e. below 450 km [37], the atmosphere is sotenuous that the flow can no longer be considered a continuum. In this scenario, the principlesthat rule the interaction of gas constituents with each other and with a body immersed in theflow change, as do the assumptions used to investigate the aerodynamic environment. The dis-criminating factor employed to distinguish between the di ff erent regimes is a dimensionless ratioknown as the Knudsen number ( K n ), which compares the order of magnitude of the molecularmean free path ( λ ) with a characteristic dimension of the flow field ( L ): K n = λ L (3)As evidenced by the definition, the Knudsen number is not strictly a flow property sinceits value is partially controlled by the reference length adopted to describe the field of motion. L is generally assumed to coincide with a significant dimension of a body in the flow, but thisassociation is not unique. Between λ and ρ there is an inverse proportional correlation, accordingto which, high values of the mean free path (and thus K n ) are usually associated with low densitylevels or gas-surface interactions occurring at nano-scale length.3hree fundamentals regimes are usually identified according to the Knudsen number. Small K n (0 < K n < .
1) are typical of the familiar continuum dynamics, where collisions between par-ticles is the prevalent mechanism of interaction. When K n → ∞ , the length travelled by theparticles before impingement with other particles in the gas mixture is considerable compared tothe characteristic flow-field dimension. In these conditions, the flow is characterised by a struc-ture in which the interaction of the gas-particles with the surface dominates over inter-particlecollisions. The distance a reflected particle travels before colliding with the free stream is of thesame order of magnitude of λ and consequently, the flow itself can be considered collisionless. Inthese conditions, it can be assumed that the presence of a body in the flow-field does not perturbthe distribution of the incident stream of particles in the vicinity of the body itself, and conse-quently, no shock waves are expected to arise. This behaviour is typical of a highly rarefied flowregime, more commonly known as free molecular flow (FMF). The majority of authors agreeto set K n >
10 as the lower limit to identify this region [9, 27, 38–42], with some variations[43]. Intermediate K n values identify the near-free molecular regime ( K n (cid:29)
1) and the complextransitional regime ( K n ∼ K n ,λ ii = λ ii L > K n ,λ ir = λ ir L >
10 (4)where λ ii is the mean free path referring to the interaction with other incident particles and λ ir is the mean free path related to the interaction with the reflected particles. According to itsdefinition in kinetic theory, the mean free path is inversely proportional to the e ff ective collisioncross-sectional area ( π d ) and the number density of the particles ( n ) [40]: λ ∝ π d n (5)where d is the radius of the sphere of influence. For the problem examined, the velocity of thesatellite through the gas is considerable and the re-emission of the particles from the surfaceis typically thermal. In these conditions, the number density of the reflected molecules cansignificantly increase, especially in proximity of the surface, possibly changing the nature of theflow locally. If the coordinate system is assumed to be fixed with the body immersed in the flow, λ ir tends to be an order of magnitude smaller than λ ii [40] and should thus be preferred for aconservative estimation of K n . For many applications, however, the free stream mean free path( λ ∞ ), defined with regards to a coordinate system fixed with the gas, is adopted: λ ∞ = √ π d n i (6)where n i is the number density of the free stream incident particles. The relation between λ ir and λ ∞ is provided by Sentman [40]: 4 ir = (cid:114) π s (cid:114) T r T i λ ∞ (7)where s is the molecular speed ratio defined later in this section, T r is the temperature of thereflected particles and T i is the temperature of the incident particles. Generally for satellites inVLEO, s > T r / T i <
1. As a consequence, λ ∞ can be considerably bigger that λ ir . Resultsshown by Walker et al. [45] suggest that the K n number referred to freestream should be in theorder of 10 to assure overall free molecular conditions. Detailed analysis of the uncertaintiesrelated to the K n number computation for orbital aerodynamics applications is provided in [40].Because of the extremely low density of the upper atmosphere, VLEO is generally charac-terised as a FMF environment. While the Knudsen number defines the degree of rarefactionof the gas, another parameter is needed to indicate how the relative magnitude of the satellite’svelocity and the most probable gas velocity a ff ects the aerodynamics. Just as the Mach numberexpresses the relationship between the body’s macroscopic velocity and the speed of sound, themolecular speed ratio indicates the ratio between the gas macroscopic velocity ( v m ) and the mostprobable molecular thermal velocity ( v t ) according to a Maxwell-Boltzmann distribution: s = v m v t = V ∞ (cid:113) RT ∞ m m (8)In Eq. 8, R is the gas constant, m m is the gas molecular mass and T ∞ is the gas kinetic temperatureof the free stream. For the orbital aerodynamics problem, the velocity of the body immersed inthe FMF is so high that the investigation concerns the motion of a body travelling at high speedsthrough a gas at rest. Consequently, in Eq. 8 the gas macroscopic velocity e ff ectively correspondsto the satellite’s velocity ( V ∞ ) and the two can be used interchangeably.At a certain altitude, variations in thermal velocity - and thus internal energy - occur withalterations in the amount of energy absorbed by the atmosphere [4, 46, 47]. In these conditions,the random thermal velocity may play a role in the determination of the induced aerodynamicforces. This behaviour is generally associated with small values of s and the FMFs are accord-ingly referred as hypothermal flows.On the contrary, when s → ∞ , the e ff ect of the bulk velocity of the particles on the aerody-namic forces estimation is considered predominant. Under these conditions, the flow is said to be hyperthermal and approximate kinetic theories ignoring the drift caused by the random thermalmotion of the particles are usually preferred. Typical values of s at VLEO altitudes are greaterthan 5. For general applications, the hyperthermal assumption is considered valid for s > F aero = m a = ρ V rel S re f C F (9)where, similarly to Eq.1, ρ is the thermospheric density, V rel is the satellite velocity with regardsto the oncoming flow, S re f is the selected reference surface and C F = [ C A , C S , C N ] T is thevector of the three aerodynamic components along the axial, the side and the normal direction,5espectively. Similarly, the resulting aerodynamic torques referenced to the centre of mass aregiven by: T aero = r PO × m a = ρ V rel S re f l re f C M (10)where C M = [ C l , C m , C n ] T is the vector of the roll, pitch and yaw momentum coe ffi cients and r PO is the position vector defining the distance between the aerodynamic centre of pressure andthe centre of mass. The magnitude of the aerodynamic forces and coe ffi cients has been estimatedin literature for bodies of di ff erent shapes making use of both analytical [41, 48–51] and numer-ical techniques [52, 53]. Both approaches have benefits and drawbacks and, ideally, the mostadvantageous strategy would be to adopt them in synergy, when permitted.Regardless of the simulation technique, the estimation of the aerodynamic coe ffi cients relieson the models employed to physically describe the underlying mechanism for GSI. C F and C M are generally computed extending the integrals of the local stress coe ffi cients ( c F ) to the surfaceexposed to the incident flow: C F = (cid:90) S c F dS (11) C M = (cid:90) S r PO × c F dS = (cid:90) S (cid:0) r P − r O (cid:1) × c F dS (12)The variety of proposed GSI models translates into a variety in c F formulations. In particular, theexpressions found in literature suggest that the aerodynamic coe ffi cients are a complex functionof a number of parameters which, once again, vary with the model adopted (Fig. 1 and 2). Acertain agreement is however found in the use of the so-called accommodation coe ffi cients, thewall temperature which is usually assumed constant, the incident gas kinetic temperature andthe molecular speed ratio s . These parameters are likely to depart from their initial value withvariations in surface contamination, composition and structure, surface thermal properties andincident particle energy and velocity. Further dependencies on geometry and velocity vectordirection are incorporated by the components of stress acting perpendicularly and tangentially tothe surface.The incident free stream, assumed to be in local Maxwellian equilibrium, interacts with thesurface transferring both energy and momentum to it. Kinetic theory-based GSI models try to de-scribe the physics underlying this phenomenon according to the contribution coming from boththe incident and re-emitted stream of particles, with major di ffi culties found in establishing a sat-isfactory mathematical model for the latter. The amount of energy and momentum exchanged isa measure of the equilibrium the impinging particles achieve with the surface before re-emission.Both phenomena are described by means of a set of average phenomenological coe ffi cients. The thermal or energy accommodation coe ffi cient , first introduced by Knudsen [54]: α T = E i − E r E i − E w = T k , i − T k , r T k , i − T w (13)describes the energy exchange, assuming that the translational, rotational and vibrational ener-gies of the particles are all a ff ected to the same degree by the interaction with the wall [43]. InEq. 13, E i and E r are the kinetic energies carried by the incident and the scattered fluxes, while E w denotes the energy that would be carried away from the surface by the scattered flux if com-plete thermal equilibrium was achieved and particles were re-emitted according to a Maxwellian6istribution corresponding to the surface temperature ( T w ). Similarly, T k , i and T k , r indicate the ki-netic temperatures of the incident and reflected streams. In accordance to what will be discussedin the following sections, it is also appropriate to introduce a partial thermal accommodationcoe ffi cient [55], whose value depends on the specific incident ( θ i ) and scattering ( θ r ) directionsselected with regards to the normal to the surface: α T , P (cid:0) θ i , θ r (cid:1) = E i (cid:0) θ i (cid:1) − E r (cid:0) θ r (cid:1) E i (cid:0) θ i (cid:1) − E w (14)To describe the momentum exchange, it is common practice to refer to the momentum coef-ficient ( σ ) [56]. Better physical correlation is usually achieved by adopting two separate accom-modation coe ffi cients to describe the normal ( σ n ) and the tangential ( σ t ) momentum exchange[43]: σ n = p i − p r p i − p w (15) σ t = τ i − τ r τ i τ w = α T , the only dif-ference being that in this case they refer to the momentum rather than the energy of the fluxes.The information of most significant value provided by α T , σ n and σ t is that the distribution ofthe re-emitted particles and velocity is deeply influenced by the degree of accommodation of theincident molecules with the surface. By referring to these quantities, two classical and extrememechanisms of interaction are identified, namely specular reflection and di ff use re-emission. Ifspecular reflection occurs without any thermal accommodation, the molecules are elastically re-flected, no thermal energy is transferred to the body and momentum exchange occurs only alongthe normal to the surface ( α T = σ n = σ t = ff use re-emission with complete thermal accommodation occurs ( α T = σ n = σ t = ffi cientenergy to generate Atomic Oxygen (AO) from the diassociation of O . The chances for the highreactive AO to reassociate to form O or O are quite low because of the very large mean freepath characterising extremely rarefied flow regimes. Because of this, AO represents the mainatmospheric constituent at VLEO altitudes and the main source of contamination and degrada-tion of surfaces exposed to the flow. E ff ects of AO interaction with polymers and metals includeoxygen erosion and inclusion, as well as formation of volatile and non-volatile reaction prod-ucts [57]. Because of the high degree of contamination of the surfaces in VLEO, most worksassume di ff use reflection with complete thermal accommodation. However, if highly accom-modated particles are likely to be predominant below 300 km [58], the same cannot be said athigher altitudes, where the atmosphere gradually becomes less and less dense, thus limiting thecontaminant adsorption to the spacecraft surfaces. According to this, Moe and Moe [58] pro-posed a Maxwellian-like model to compute the drag coe ffi cient. The latter uses a modified form7 igure 1: GSI models compared: parameters of dependence for scattering distributions and aerodynamic coe ffi cients.Black dots are used to identify parameters explicitly declared in the formulations, while white rhombus indicate implicitparameters of dependence. Parameters are grouped according to the following families: [1] flow angles; [2] energyparameters; [3] accommodation coe ffi cients; [4] beam shape parameters; [5] velocities; [6] interaction parameters; [7]mass parameters. igure 2: Comparison between GSI models assumptions.Figure 3: Mechanisms of re-emission for specular reflection without thermal accommodation (left) and di ff use reflectionwith complete thermal accommodation (right). igure 4: Comparison between Sentman’s [40], Cook’s [61] and Schaaf & Chambre’s [43] model for hypothermal flows( s =
6) and hyperthermal flows ( s = c p and c τ predicted by the models are shown, assuming α T = σ n = σ t = T w =
300 K and T ∞ = of Sentman’s model to compute the contribution associated with the 0 < σ < ff usely re-emitted. Schamberg’s model [59] is instead used in combinationwith Goodman’s accommodation coe ffi cient [60] to address the input coming from the (1 − σ )fraction that is quasi-specularly reflected. Schamberg’s [59] and Sentman’s [40] are probably themost popular models adopted to perform the estimation of aerodynamic properties. However,like many other GSI models, they rely on a specific set of assumptions that restrict their rangeof applicability. Schamberg’s quasi-specular model assumes hyperthermal impinging FMF anduniform scattering speed along all directions. The hyperthermal approximation, conserved byCook in a re-adaption of the model [61], provides valid results for applications in VLEO. How-ever some care must be taken since, as more rigorously discussed in [40], neglecting the particlethermal velocity may introduce some errors for small angles of attack even for s > ff ect of neglecting particle thermal velocity is more visible when the aerodynamic co-10 ffi cients are expressed in terms of normal (cid:16) c p (cid:17) and shear ( c τ ) stress components: C D ≈ c p cos θ + c τ sin θ (17) C L ≈ c p sin θ − c τ cos θ (18)Fig. 4 shows the aerodynamic behaviour of a flat plate immersed in free molecular flow, accord-ing to a hyperthermal model (Cook [61]) and non-hyperthermal models (Schaaf & Chambre [43],Sentman [40]). For varying molecular speed ratios, variations of c τ are noticeable for θ ∼ ◦ ,and thus grazing angles of attack. The predicted c p values vary as well, showing noticeable de-viation and disagreement between hypothermal and hyperthermal models at low s values. Onthe other hand, Sentman’s model considers molecules random thermal motion, but it is built onthe assumption of complete di ff use reflection. These models will not be discussed in detail here,since a comprehensive review can be found in [9, 27]. However, their fundamental assumptionsare summarised in Fig. 1 and 2.
2. Scattering-kernel theory based GSI models
Scattering-kernel models, as we will refer to them in this section, are kinetic models built on astatistical approach. Correlation between the incident and reflected distributions of the particlesis achieved through a proper definition of the boundary conditions for the collisionless Boltz-mann equation in the rarefied gas regime. The correct formulation of these is of fundamentalimportance since boundary conditions describe the mechanisms that rule the interaction betweengas and solid particles. When boundary conditions are written within the frame of the scattering-kernel theory, the problem consists in finding the most suitable expression for the scatteringkernel K ( x , t , ξ i → ξ r ) to reproduce the interaction phenomena occuring at the surface under theassumptions considered.Since an accurate knowledge of the dynamics and the thermodynamics of the interaction isnot achievable, a quantity P ( ξ i → ξ r , x i → x r , t i → t r ) can be introduced to describe the densityof probability that a gas particle impacting on a point x i located on the surface, at a certain time t i with an incident velocity corresponding to ξ i is reflected at a generally di ff erent location x r attime t r with a velocity ξ r (cid:44) ξ i [62]. Assuming that the sitting time on the surface is low enough( t i (cid:39) t r = t → x i (cid:39) x r = x ) such that no adsorption / desorption or di ff usion phenomena need tobe addressed [62]: P ( ξ i → ξ r , x i → x r , t i → t r ) = K ( ξ i → ξ r ) (19)where the incident and reflected molecular velocity vectors more generally consist of two tan-gential ( (cid:2) ξ i , t (cid:48) , ξ i , t (cid:48)(cid:48) (cid:3) , (cid:2) ξ r , t (cid:48) , ξ r , t (cid:48)(cid:48) (cid:3) ) and one normal ( ξ i , n < ξ r , n >
0) component. Proposed classesof K ( ξ i → ξ r ) not only need to correctly capture the phenomena occuring in the proximity of thewall through a proper correlation of the incident (cid:0) f i (cid:0) ξ i (cid:1)(cid:1) and reflected (cid:0) f r (cid:0) ξ r (cid:1)(cid:1) velocity distribu-tions: ξ r , n f r (cid:0) ξ r (cid:1) = (cid:90) ξ i , n < K (cid:0) ξ i → ξ r (cid:1) | ξ i , n | f i (cid:0) ξ i (cid:1) d ξ i (20)but also need to satisfy some specific conditions. The non-negativity condition [63]:11 ( ξ i → ξ r ) ≥ R . The normalisationcondition [63]: (cid:90) ξ r , n K ( ξ i → ξ r ) d ξ r = f ( ξ i ) | ξ i , n | K ( ξ i → ξ r ) = f ( ξ r ) | ξ r , n | K ( − ξ r → − ξ i ) (23)is a balance equation according to which, for a gas-surface system in thermodynamic equilib-rium, a correspondence can be established between 1) the scattering path of the particles and 2)the path that those particles would hypothetically follow if they travelled with velocities that areopposite to those considered in the same time interval [64]. This constraint stems by writingthe boundary condition as a function of the temperature of the wall. In Eq. 23, f indicates theMaxwellian velocity distribution function corresponding to T w . In Maxwell’s model [65] the behaviour of the reflecting surface is described by the linearcombination of the two classical scattering characteristics mentioned in the previous paragraph.According to this, a fraction of incident particles, identifiable with the accommodation coe ffi cient σ , is assumed to be trapped by a perfectly absorbing wall. After achieving complete accommoda-tion, the particles are re-emitted from the surface with a Maxwellian velocity distribution typicalof a gas at rest and in thermal equilibrium with the wall ( T g = T w ). The remaining fraction (cid:0) − σ (cid:1) of the incident gas particles collides on an ideally smooth and elastic surface so that, afterthe collision, the momentum exchange occurs just along the normal direction and the reflectedgas lies along the specular direction. The scattering kernel for this model is accordingly built asthe sum of the scattering kernels for specular reflection and di ff use re-emission with completethermal accommodation [66, 67]: K M (cid:0) ξ i → ξ r (cid:1) = (cid:0) − σ (cid:1) δ (cid:0) ξ i − ξ r , specular (cid:1) + σ f (cid:0) ξ r (cid:1) | ξ r , n | (24)where δ is the Dirac delta function. This isotropic scattering kernel satisfies the physical con-straints discussed and, because of its simplicity, has found substantial success in DSMC imple-mentations. According to the model however, scattering cosine distributions presenting peaks inproximity of the the specular direction (Figure 5) should be expected. As discussed in Section4, experimental results show a more complex behaviour which is not predictable by the linearcombination adopted by Maxwell, thus limiting the range of applicability of the model.12 igure 5: Schematic representation of the polar plot distribution predicted by the Maxwell model [56]. The NHS model, first proposed by Nocilla [68, 69] and refined by Hurlbut and Sherman [55]postulates a ”drifting / shifted Maxwellian” law for the reflected particles distribution: f r (cid:0) ξ (cid:1) = n r (cid:0) π RT r (cid:1) / exp (cid:34) − (cid:0) ξ − v r (cid:1) RT r (cid:35) (25)where n r is the scattering molecules density, T r is the temperature associated with the scattereddistribution, and v r is the macroscopic or bulk velocity (drift) of the reflected particles. Ac-cording to Equation 25, the reflected flux is described by v r , the scattering angle δ r = ◦ − θ r determined by the direction of v r with regards to the tangent to the surface, and T r which ingeneral can be di ff erent from T s . In the original form proposed by Nocilla, the model containsthe complementary cases of di ff use scattering ( v r =
0) and specular reflection ( s r = s i , v r = v i , δ r = δ i ). However, the mathematical formulation inherently su ff ers from the lack of connectionbetween the incident and reflected velocity distributions. In this regard, Hurlbut and Sherman[55] introduced a drifting Maxwellian velocity distribution for the incident particles: f i (cid:0) ξ (cid:1) = n i (cid:0) π RT i (cid:1) / exp (cid:34) − (cid:0) ξ − v i (cid:1) RT i (cid:35) (26)and related the two distributions by determining n r from the equivalence of the incoming andreflected number fluxes in stationary conditions. A model reformulation was also proposed toreduce the di ffi culties encountered in matching the experimental results with the model, whichrequires an accurate determination of T r . A partial thermal accommodation coe ffi cient α T , P (cid:0) θ i (cid:1) ∼ α T , P (cid:0) δ i (cid:1) , avaraged over all the possible scattering directions and defined for a specific angle ofincidence δ i , was accordingly introduced [55]: α T , P (cid:0) δ i (cid:1) = E i (cid:0) δ i (cid:1) − E r E i (cid:0) δ i (cid:1) − E w (27)13nd analytic expressions for the computation of the aerodynamic drag and lift coe ffi cients weredetermined. For a given angle of incidence of the flow, the aerodynamic coe ffi cients are built as afunction of v r (or s r ), δ r and α T , P (cid:0) δ i (cid:1) . Cercignani and Lampis [70] provided a reformulation of theNHS model in the context of the scattering kernel theory, following some suggestions alreadypresent in Nocilla’s original proposal. Corrections to this model were also applied so that theproposed kernel could satisfy the normalisation (Eq. 22) and detailed balance conditions (Eq. 23)[71]. Despite these improvements and some early applications of the model for the computationof the aerodynamic coe ffi cients in FMF conditions [72], further implementations did not findmuch success. However, the good agreement achieved with some gas-beam experimental results[55] for clean surfaces made the NHS model a fundamental starting point for advances in thestudy of gas-surface interaction. The Cercignani-Lampis [62] model (CL) is one of the most successful kernel-based repre-sentations of the gas particles-surface interaction as it is described by experimental results. Theexpression for the scattering kernel is obtained assuming no adsorption and independent interac-tion of each gas particle with the surface [62]: K CL (cid:0) ξ i → ξ r (cid:1) = α n σ t (cid:0) − σ t (cid:1) exp (cid:20) α n − α n (cid:0) ξ r , n + ξ i , n (cid:1) + − (cid:0) − σ t (cid:1) σ t (cid:0) − σ t (cid:1) (cid:0) ξ r , t + ξ i , t (cid:1) ++ (cid:0) − σ t (cid:1) σ t (cid:0) − σ t (cid:1) (cid:0) ξ i , t · ξ r , t (cid:1)(cid:21) I (cid:20) (cid:0) − α n (cid:1) / α n ξ r , n ξ i , n (cid:21) (28)where for both the incident and reflected velocity, ξ j , t = (cid:113) ξ j , t (cid:48) + ξ j , t (cid:48)(cid:48) and I is the modified BesselFunction of the first kind and zeroth order. The density of probability described by Eq. 28 is givenby the contributions coming from the variation of the three components of the velocity vector.Since the tangential and the normal components can be treated separately in the model, theirindividual scattering kernels can accordingly be derived. Following the formulation proposed byLord [73] for isotropic surfaces, the scattering kernel for the normal component of velocity canbe written: K CLL (cid:0) ξ i , n → ξ r , n (cid:1) = ξ r , n α n I (cid:0) − α n (cid:1) / ξ r , n ξ i , n α n × exp (cid:20) − ξ r , n + (cid:0) − α n (cid:1) ξ i , n α n (cid:21) (29)for which the reciprocity and normalisation conditions take the form: | ξ i , n | exp (cid:2) − ξ i , n (cid:3) K CLL (cid:0) ξ i , n → ξ r , n (cid:1) = ξ r , n exp (cid:2) − ξ r , n (cid:3) K CLL (cid:0) − ξ r , n → − ξ i , n (cid:1) (30) (cid:90) ∞ K CLL (cid:0) ξ i , n → ξ r , n (cid:1) d ξ r , n = ffi cient so that, following surface isotropy, the expressions of the scattering kernels for the twotangential components of velocity are in the form of: K CLL (cid:0) ξ i , t → ξ r , t (cid:1) = (cid:112) πσ t (cid:0) − σ t (cid:1) × exp (cid:26) − (cid:2) ξ r , t − (cid:0) − σ t (cid:1) ξ i , t (cid:3) σ t (cid:0) − σ t (cid:1) (cid:27) (32)which satisfies: exp (cid:2) − ξ i , t (cid:3) K CLL (cid:0) ξ i , t → ξ r , t (cid:1) = exp (cid:2) − ξ r , t (cid:3) K CLL (cid:0) − ξ r , t → − ξ i , t (cid:1) (33) (cid:90) ∞−∞ K CLL (cid:0) ξ i , t → ξ r , t (cid:1) d ξ r , t = ffi cient ( α n ) and the tangential momentumaccommodation coe ffi cient ( σ t ). Lord contributed significantly to the success of the CL modelwhilst adapting it for DSMC implementation [74] and further extended its validity to cases ex-cluded by the original model, among which di ff use re-emission with incomplete accommodation[73, 75] . Because of this, it is generally preferred to refer to the model as the Cercignani-Lampis-Lord model (CLL).A possible implementation of the CLL model in closed-form solutions was proposed byWalker et al. [45]. The authors suggested modified expressions of the Schaaf and Chambre[41, 43] (S&C) closed form equations with the CLL model. The attempt is rather di ffi cult since,among other parameters (Fig. 1), the S&C closed-form solutions are written as a function of σ n and σ t . While an immediate relation can be found between the tangential momentum accommo-dation coe ffi cient and the tangential energy accommodation coe ffi cient ( α t ): α t = σ t (cid:0) − σ t (cid:1) (35)the same cannot really be said for α n and σ n . Approximate analytic σ n − α n relations, writtenas a function of four best-fit parameters, were found adopting a least squares error approach andsensitivity analysis. Ranges of variation were selected for some meaningful parameters abouttheir nominal values. In this way the agreement between the σ n − α n relation and the expected C D values could be evaluated for the nominal conditions and over the range of variation of theselected parameters. These last were identified with the bulk velocity of the particles, the freestream temperature, the surface temperature, the normal thermal accommodation coe ffi cient andthe tangential momentum accommodation coe ffi cient. Good correlation between the computed C D and the values provided by the CLL model implemented in DSMC Analysis Code was seenfor the modified S&C closed-form solutions written as a function of the derived σ n − α n laws.However, the set of values to be chosen for the best-fit parameters is not constant but varieswith the gas species considered, the type of body impinged and, in the case of lighter molecularspecies, the α n range assumed. Values suggested by Walker et al. for representetive molecularspecies and body shapes could be found in the original paper from the authors [45]. Biggeruncertainties are found in the case of He and H for value of α n close to unity.15 . Physical GSI Models Physical GSI models take advantage of experimental results to describe how the thermal mo-tion of the surface influences the scattering dynamics of the impinging gas-beam. These modelsare thus based on assumptions regarding the surface interaction potentials, the surface morpho-logical structure and the surface elasticity / sti ff ness characteristics. Among the vast number ofmodels present in literature, special attention will be devoted to the simple quasi one-dimensionalHard Cube model and its most successful expansions: the Soft Cube model and the Washboardmodel. Two and three-dimensional lattice models [76–79] will be neglected in this review. Theselast are generally characterised by a more complex implementation which leads to higher accu-racy and also computational time; factors potentially limiting the range of applicability of thesemodels for the context of this paper. Moreover, if the increase in complexity is justified inthe frame of pure gas-surface interaction, the same might not be true for orbital aerodynamicsengineering applications. The number and range of uncertainties a ff ecting the problem of aero-dynamic forces and torques estimation in VLEO is quite high. Because of this, the increasedlevel of complexity is likely to be unjustifiable against the numerous sources of errors observed. The Hard Cube Model, as proposed in its earlier form by Goodman [80], has found successwith Logan’s and Stickney’s [81] formulation. Despite its inherent simplicity, resulting from theassumptions adopted in its development, the model is able to qualitatively reproduce the exper-imental lobal scattering typical of clean and polished surfaces. The model assumes the surfaceto be perfectly smooth and the gas particles and surface atoms involved in the interaction to beideally elastic and rigid. The dynamics of the collision is simplified by assuming that each gasparticle interacts solely with a surface atom represented as an isolated cube in the lattice, so thatany impact of the surface structure on the scattering properties is neglected. During the colli-sion, the gas particle and the surface atom interact as free particles so that a one-dimensionalimpulsive-repulsive potential, with no attractive well, can be conjectured. In this way, the impactof interaction times on the collision mechanism can be ignored with benefits in terms of sim-plicity and with only a partial loss of accuracy. The cubes comprising the surface are orientedso that one of their four faces lies in the direction parallel to the surface contour and they arecharacterised by an initial Maxwellian normal velocity distribution determined by the surfacetemperature. The momentum exchange is assumed to be due uniquely to the normal compo-nent of the gas particle velocity ( v n ) as the tangential component ( v t ) is preserved by the surfaceproperties (Figure 6). The model predicts a quasi-specular re-emission both above and belowthe specular range with each scattering angle θ r being determined by a unique value of v r , n for agiven v i , t = v r , t .Along with the numerical formulation, an analytical approach in which mean velocity valuesare adopted instead of velocity distributions for both the gas particles and the surface atoms wasproposed by Logan and Stickney [81]. According to this approach, closed form solutions can bederived and the parameters on which the interaction depends can be more easily identified. Theflat surface assumption allows the restriction of the analysis to the plane identified by the surfacenormal and the incident velocity vector, so that: θ r = cot − (cid:34) cot ( θ i ) (cid:32) − µ + µ + µ π (1 + µ ) T w T g cos ( θ i ) (cid:33)(cid:35) (36)where µ , the gas particle-surface atom mass ratio, is restricted to vary in the following range:16 igure 6: Hard Cube Model, reproduced from [81]. < µ = m g m s <
13 (37)and T g is the gas-beam source temperature. Constraints on the possible values of µ result fromfurther assuming that the gas particle experiences a single interaction with the surface atomconsidered. According to equation (36), the scattering direction depends on the mass ratio µ , theincidence angle and the surface-to-gas temperature ratio. Simplicity and ease in implementationare however obtained at the price of a general loss in accuracy in describing the experimentalresults compared to the extended model described in [81], for which numerical simulation isneeded.Some expansions of the HC model have been proposed, the most successful one discussedlater in this section. Hurst et al. [82] and Nichols et al. [83, 84] modified the model to capturerotational dynamics of elliptically-shaped diatomic molecules scattered from the surface. Doll[85] addressed the rotational dynamics modelling diatomic homonuclear molecules as rigid ro-tors with motion restricted to a single plane. The importance for this model of multiple collisionswith the surface atom arising from the rotational state were also discussed. Trapping phenomenawere addressed by Weinberg and Merrill [86]. Trilling and Hurkmans [87] introduced an attrac-tive long-range Coulomb potential, an exponential short-range repulsive potential and modifiedthe surface geometry treating atoms as ”spherical caps” rather then cubes. Sitz et al. [88] ex-panded the HC model to qualitatively describe momentum orientation in the scatterring of N from smooth Ag(111), introducing a frictional force along the surface tangential direction. Theadditional level of complexity characterising these models seems inappropriate for the applica-tions for which this review paper is intended and, because of this, they will not be discussed morethoroughly in the following sections. 17 igure 7: The Soft Cube Model, reproduced from [89]. The Soft Cube Model proposed by Logan and Keck in 1968 [89] owes its name to the intro-duction of a more realistic ”soft” potential to capture the physics of the gas-surface interaction.The atoms that comprise the flat surface and take part in the collision are assumed to behavelike independent cubes linked to the underlying lattice by means of single linear springs (Fig-ure 7). Surface atoms are thus regarded as oscillators characterised by a natural frequency ω and a Maxwellian energy distribution corresponding to the surface temperature T w . Similarlyto the HC model, the interaction with a gas particle involves a single cube in the lattice andthe energy exchange, responsible for the accommodation coe ffi cient value, is due solely to thevariation of the normal component of velocity after the collision ( v i , t = v r , t ). The interactionis however more realistically captured assuming non-negligible collision times, described by anon-impulsive interaction potential consisting of two components. In addition to a repulsiveexponential potential, which substitutes the impulsive repulsive interaction assumed in the HCmodel, an attractive long-range square-well potential component is introduced. The model canbe eventually employed to obtain an estimate of the fraction of particles that remain trapped onthe surface after the collision and depart from it after achieving su ffi cient energy.Comparison with experimental results is obtained by properly selecting the value of threemodifiable parameters: 1) the potential well-depth W , 2) the interaction range b and 3) the oscil-lator frequency ω , whose value is assumed to be given by the Debye temperature ( Θ D ). Combi-nations of b and W that reproduce, with satisfactory agreement, the experimental data referred toa selection of gas-surface systems can be found in the original paper by Logan and Keck [89]. Like the Soft Cube Model, the Washboard model [90] can be regarded as an attempt toimprove the Hard Cube model agreement with the experimental results. Compared to the otherGSI models discussed so far, the Washboard model has the advantage of addressing the e ff ect ofsurface corrugation on the scattering properties while preserving relative clarity and simplicity.18 igure 8: The washboard model, reproduced from [90]. The surface contour is simplified assuming a sinusoidal profile applied exclusively in one di-rection, thus making the model appropriate for bidimensional but not for out-of-plane scatteringevaluation. Similarly to the HC model, the cube with which the colliding gas particle interactsis oriented along the surface contour and its velocity is determined by a Maxwellian distributionat the surface temperature. Because of the surface corrugation, the cubes are tilted with regardsto the normal to the flat surface so that, for any impact point, a local normal and tangential di-rections can be identified (Figure 8). The maximum deviation of the local normal from the flatsurface normal direction is measured by the corrugation strength parameter ( Ω C ). The introduc-tion of this parameter allows the model to adapt to di ff erent levels of surface corrugation, thusproviding good qualitative agreement with experimental results ranging from smooth to roughsurfaces. The attractive potential well W produces refraction in the gas particle initial trajectory,thus varying its normal and tangential component of velocity. Even in this case the nature ofthe interaction is impulsive so that in the local normal-tangential reference system, the tangentialmomentum is unchanged and the energy exchange is determined solely by the normal momen-tum variation. As a consquence, in the xz flat surface reference system the tangential momentumis not conserved and the strict assumption characterising both the Hard and the Soft Cube modelis accordingly removed.Analytic expressions for the angular scattering velocity and kinetic energy distributions wereprovided for small surface corrugations. The parameters on which these depend are the corruga-tion strength coe ffi cient Ω C , the incident angle θ i , the incident kinetic energy E i , the mass ratio µ , the potential well W and the surface temperature T w . Eventually, the trapping probability canalso be addressed. Further extensions of the washboard model include the works of Yan et al.[91] and Liang et al. [92].
4. Comparison of GSI Models with Gas-Beam Experiments
Molecular beam experiments are a useful means of gaining information concerning the en-ergy exchange at the surface and the wall-gas system characteristics at atomic scale. For appro-priate selection of incident energy distributions, the scattering of neutral molecules on a target19s representative of orbital conditions and can thus be used to investigate the aerodynamic be-haviour of specific materials in the context of space applications. The literature covering thetopic is vast and this section does not claim to be exhaustive, since such an activity would re-quire a specific review e ff ort. The objective is thus to provide an overall picture of the subject,focusing the attention on the physical regimes that are significant for the purpose of this paper,thus addressing, wherever possible, the points of strength and the limits of the models previ-ously discussed in the context of VLEO aerodynamics. At VLEO altitudes, AO is the dominantconstituent of the residual atmosphere, with atoms impacting on the exposed surfaces with anaverage velocity of ∼ . km s − corresponding to a Maxwellian mean incident energy distri-bution of ∼ eV . According to this, attention will be devoted especially to results referring tomolecular beam scattering of monoatomic species from targets in a variety of conditions. Studiesanalysing diatomic and polyatomic beams scattering are numerous but their results are typicallymore di ffi cult to interpret: the interaction depends on both the translational and internal ener-gies of the molecule considered and on the local aspect of the interaction potential. However,results for N and O scattering from Ag(111) reported by Asada et al. [93] show mean veloc-ity and mean energy distributions with scattering angle which are similar to those obtained formonoatomic molecules. Similarly, analysis of adsorption and desorption rates, which are im-portant especially for heavier molecular species, requires correlation between the characteristicsof the system interaction potential and the sticking probability. These, in turn, vary with initialrotational and translational energies, binding energies, orientation of the molecules, incidenceangle and surface temperature [94]. Because of this, the spatial distributions obtained are theresult of a complex mechanism of interaction [95, 96] and more advanced numerical tecnhiquesinvolving molecular dynamics or binary collision approximation are required. Extensive reviewson the topic are however available in [97–102] with some more dated results reported in [103].The scattering behaviours observed are not constant as they tend to change substantially withthe system considered and, in particular, with the ratio between the mass of the gas particles andthe surface atoms, the range of interaction, the energy / temperature of the incident beam withregards to the surface temperature, the molecular or atomic species involved in the experiment,the presence of adsorbents on the target surface, the morphology of the surface considered despitethe level of roughness and the relative position and orientation of the gas particles and surfaceatoms [97]. At the time of writing, a comprehensive theory capable of capturing each possiblescenario is not available such that multiple models, suitable for specific physical regimes, areadopted instead. For light particles and low incidence energies [98] compared to solid maximumphonon energy, the interaction is predominatly elastic. In these conditions, no energy transferoccurs in the system and quantum mechanical phenomena, such as di ff raction, are expected tobe predominant [97].As gas particles mass and incident energy increases, the collisions become more inelasticin nature and classic theory is adequate enough to reproduce the scattering behaviour. For thisphysical range, whether particles gain or transfer energy to the surface depends on the relativemagnitude of E i and T w . Generally three mechanisms of interaction are identified: 1) Singlegas-surface collision with moderate net energy exchange; 2) Multiple gas-surface collisions withno adsorption and delayed scattering; 3) Multiple gas-surface collisions with adsorption to thesurface and eventual desorption in the scattered gas. The latter interaction mode is typical ofhighly contaminated surfaces [104–106]: in these conditions, the adsorbed particles have time toreach equilibrium with the surface and they are scattered according to a cosine distribution with θ r and a Maxwellian translational velocity distribution corresponding to the wall temperature.The majority of works published in literature, however, refer to the first two interaction scenarios,20ypically observed in scattering from clean flat surfaces in Ultra High Vacuum (UHV) conditions.In this case, lobal re-emission distributions characterised by a predominant asymmetrical quasi-specular component are observed for varying surface properties, incident particles energies andwall temperature [93, 106–110, 110, 111]. The generally small amount of particles that undergomultiple collisions before being scattered in the gas-phase determines the width of the lobe ofdistribution and seems to be susceptible to the morphology of the surface considered, despite thelevel of roughness. Wider angular distributions were obtained for smoother surfaces even whenhigher incident energies were employed [107]. This seems to suggest that the interactions withinatoms in the same surface layer or adjacent layers may play a role in determining the scatteringcharacteristic.In the inelastic scattering domain, however, di ff erent scattering behaviours and thus di ff erenttrends are expected according to the incident energy and the interaction radius for the systemconsidered [112]. As highlighted by Goodman in [97], for gas-surface systems that are notcharacteristed by a strong periodicity in the interaction potential, low values of incident kineticenergy ( E i < k B T w ) and large interaction distances define the so-called thermal scattering regime. In these conditions, the impinging gas-particles can not see the surface corrugationand the surface appears flat and smooth. During the interaction, the tangential component ofmomentum is generally conserved and the scattering dynamics are dominated by the surfacethermal motion in the direction normal to the surface. Thermal scattering studies [106, 109, 113–116] show the following common features for the angular distributions (Fig. 9):1. ∂θ r , max /∂ T w ≤
0: the scattering angle corresponding to the peak in the distribution slightlymoves towards the normal to the surface with increasing T w [117]. Moreover, at highvalues of T w , for which the surface appears to be free of adsorbents, the width of thedistribution experiences a slight increase with decreasing T i / T w [106, 113]. Higher T w also induces lower scattering intensity at the peak of the distribution [113, 116]. Someauthors, however, obtained the opposite trend for the scattering of Ar [118], Xe [116, 118]and Kr [113, 118] on various metal surfaces at T i = T amb . Generally speaking, high walltemperatures are e ffi cient in reducing the sticking probability and the time required for theinteraction, thus preventing complete accommodation;2. ∂θ r , max /∂θ i ≥
0: the scattering angle corresponding to the peak in the distribution movestowards the surface tangent as the incidence angle increases [81, 106, 114, 115]. Moreover, ∂θ r , max /∂ m g ≤
0, i.e. the scattering angle for which the distribution presents a peak movestowards the normal to the surface as the mass of the incident gas atom increases [81, 106,119];As the incident kinetic energy of the beam increases (within the limits of the thermal scatter-ing domain), the e ff ect of surface thermal vibration on the angular scattering distribution becomesless dominant. As a consequence, the lobal distribution becomes narrower and more symmetricalin shape, the scattered intensity at the peak increases [113] and θ r , max generally moves towardsthe tangent to the surface [108, 112, 119, 120] (Figure 10, top right). When this condition isobserved, the re-emission distribution is said to be superspecular . The e ff ect seems to be more The interaction distance is defined here following the definition of the non-dimensional radius parameter R providedby Goodman in [97]. The distance of interaction is thus defined as the ratio between two quantities: the closest distancethat separates the centre of the impinging atom from the centre of the surface atom during the collision and the criticalvalue of this distance for the gas-surface system considered. When the critical distance value is achieved the impingingparticle enters the surface. igure 9: Variation of θ r , max - the angle at which the peak of the distribution occurs - in the thermal scattering regime.Behaviours observed with increasing T w (top left), θ i (top right) and m g (bottom) are illustrated. noticeable for nearly grazing angles of incidence [108]. With regards to translational energydistributions, when the conservation of the tangential momentum is observed, the relative ratiobetween the mean final and incident energies varies according to the parallel momentum conser-vation curve: E r E i = sin θ i sin θ r (38)The characteristics mentioned above are well described by the cube models [117] reviewed inthe previous section of this paper, because of the inherent assumptions on which these modelsare built. Better agreement, as expected, is found with the SC [89, 105, 106] rather than withthe HC model [108, 112, 113, 116, 117, 121], not only because of the more realistic gas-surfaceinteraction potential assumed, but also because adjustments can be made through the parameter W for the system considered. When the attractive well W dominates the dynamics, the repulsivepotential assumption loses accuracy and the HC model fails in describing trapping and partialaccommodation to the surface. The limitations imposed on the gas particle-surface atom massratio in the HC model are likely to make the model unsuitable for addressing atomic oxygenscattering from most surfaces. Moreover, comparison of the potentialities of these two mod-els against experimental data is possible just in the incident scattering plane. For out of planescattering considerations, techniques addressing surface corrugation in more than one dimensionneed to be employed. While Maxwell’s model fails in reproducing the petal-shaped angular dis-tribution observed in these experiments, it appears that its theoretical apparatus and simplifiedassumptions are su ffi cient to describe some re-emission polar plots showing a small nearly spec-ular and a large di ff use component [104, 107]. Results provided by Mehta et al. [108] show thatwhen the CLL model is adopted some di ffi culties are encountered in the attempt of selecting the22 igure 10: Transition from thermal to structure scattering with increasing values of E i . proper combination of α n and σ t to reproduce the experimental conditions. Excellent predic-tions of the position of the peak ( θ r , max ) and of the dispersion of the scattering distribution, forselected values of the accommodation coe ffi cients, seem to exclude an accurate representation ofthe experimental E r / E i , and viceversa.As the kinetic incident energy of the beam increases [117, 122, 123] with regards to the sur-face atoms thermal energy ( E i > k B T w ) and the radius of interaction reduces, transition to the structure scattering regime is experienced: in this case, the surface roughness sensed by the im-pinging particles is noticeable because of the increased power of penetration. The interaction isno longer dominated by the surface thermal motion but by the surface corrugation. Because ofthe multiple collissions experienced by the particles with the rough surface, the angular distribu-tion in this regime becomes wider in shape, the value of the peak scattered intensity decreasesand a shift from superspecular to directions closer to the specular range for θ r , max is observed[115, 124]. The qualitative re-emission behaviour expected for increasing value of E i in the tran-sition from the thermal to the structure regime are reproduced in Figure 10. In contrast to thethermal regime, the energy of the scattered beam ( E r ) increases with increasing θ r [117, 125].Cube models relying on the flat surface approximation, are unable to reproduce this scenarioand agreement is rather found with the Hard Spheres model [126], the Washboard model [120],and, in general, more complex models. Accordance of the HC model with some gas-beam datareferred to hyperthermal E i seems fortuitous [127] and, according to the authors, attributable tothe morphology of the gas-surface system considered. An interesting scattering phenomenonobserved in the structure regime is rainbow scattering: when this process occurs the typicallywide spatial distribution is characterised by two peak lobes corresponding to di ff erent intensitiesin the flux measurements. This phenomenon was however, predominatly observed on the scat-tering of rare gases from LiF surfaces [128, 129], with some evidence on systems composed bymetal surfaces [120, 130]. More details on the topic can be found in [97, 98]. The appearanceof rainbow scattering e ff ects seems to be associated with surface corrugation. In this regard, theWashboard model o ff ers a relatively simple formulation able of capturing more complicated re-23mission mechanisms for which the HC and SC models are not suited. Moreover, the Washboardmodel appears to be more e ff ective in describing gas particle attraction and surface penetration aswell as scattering characteristic from softer surfaces [121], especially if compared with the HCmodel. These features are particularly useful when studying gas-surface interaction specificallyfor orbital aerodynamic applications. The interaction of most materials with the orbital environ-ment is likely to cause variations in the static gas-surface potential corrugation [57] and, morein general, degradation of material performances with time. The lattice structure is subjected tobecome rougher as AO exerts its erosion action on the ram surfaces. As a consequence, materialswith promising aerodynamic performances (i.e. near-specular re-emissions) might experiencevariations in the expected scattering behaviour within a mission lifetime. In this scenario, theprinciples on which the Washboard model is built could prove useful to address materials degra-dation and performance variation.
5. Conclusions
Renewed interest in small satellites missions in the lower region of the Earth’s atmospheredemands new aerodynamic technologies capable of taking advantage of the environment in LEO.Aerodynamic performances are dependent on the mechanisms that rule the gas-solid interaction,but great uncertainties are associated with the physical processes occurring at the wall in rarefiedand extremely rarefied regimes. Gas-beam experimental results suggest that reality might bemore complex than that described by classical theoretical kinetic models. The development of anew generation of aerodynamic materials may therefore require more accurate predictions. Thenumber of uncertainties in the system behaviour reflects a vast production of models in literature:di ffi culties however arise in the attempt of combining e ffi ciency with simplicity. In the previoussections popular models which have obtained considerable success for their immediacy and abil-ity to predict scattering behaviours have been reviewed. These models were broadly associatedwith two principal families according to some common features. Scattering-kernel theory basedGSI models are built on a statistical approach, while physical GSI models provide a simple toolsto describe the complex physical interaction mechanisms observed at the wall. Wherever possi-ble, their appropriateness has been discussed against relevant gas-beam experimental results forthe problem considered and physical ranges of application have been identified.The joined e ff ort of several authors has resulted in remarkable improvements in the under-standing of the phenomena involved in the observed re-emissions. The problem is howevercomplex and multidisciplinary in nature. Despite the challenges that remain, they represent astarting point for future developments. At the moment of writing, it appears that an easy-to-implement model applicable to di ff erent scattering regimes and to di ff erent gas-solid systems isstill to be defined. Similarly, a simple analytical model capable of a more accurate quantitativedescription of the behaviours of both clean and contaminated surfaces might be desirable. In thisregard, the level of technological advancement achieved by gas-beam facilities seems adequateto support a more critical analysis of the models developed so far. A more scrupulous exami-nation of the approximations on which these rely may help in identifying the points of strengthof each model and possibly expand their range of applicability. There might also be a chanceto identify with more accuracy the re-emission patterns that an improved GSI model should beable to describe. A possible strategy may include enriching the proposed scattering-kernels forGSI with some more realistical assumptions regarding adsorption and desorption phenomena.Further comparison of theoretical models addressing surface corrugation with a broader rangeof experimental data might be helpful as well. Due to the wide amount of results concerning24cattering from clean surfaces, the majority of works seem to focus mainly on comparison withthe simpler HC and SC models. The Washboard model appears to be privileged in the study ofrainbow scattering from extremely corrugated surfaces. However, using this model against datareferring to less corrugated surfaces might facilitate the understanding of the GSI problem.Some adaptability characteristics are especially desired when VLEO aerodynamic applica-tions are discussed. The growing interest in Earth-observation missions at these altitudes haspaved the way for the study of aerodynamic materials promoting nearly specular reflection. How-ever, the results obtained in a controlled facility may be a ff ected by considerable alterations in thereal thermospheric environment. A robust GSI model capable of providing satisfactory agree-ment for a range of interaction performance can facilitate the task of designing aerodynamicfeatures for VLEO satellites. As a consequence, increase in the reliability of the aerodynamiccontrol manoeuvres proposed in literature is expected. End-of-life tasks, satellite geometry andABEP system design are likely to benefit from any knowledge improvement as well. This task,although di ffi cult, seems to be easier to achieve in the context of orbital aerodynamics. The re-quirements imposed on the accuracy of the model are more relaxed than those expected in themore general frame of gas-surface interaction science. This is a result of the considerable numberof additional uncertainties that a ff ect the estimation of the aerodynamic forces and torques. Aless accurate but still e ff ective model may therefore adequately serve the purpose of describing arange of scattering behaviour or, equivalently, aerodynamic performance.
6. Acknowledgements
The DISCOVERER project has received funding from the European Union’s Horizon 2020research and innovation programme under grant agreement No 737183. Disclaimer: This publi-cation reflects only the views of the authors. The European Commission is not liable for any usethat may be made of the information contained therein.
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