gas-surface interaction database and modelling

GAS-SURFACE INTERACTION DATABASE AND MODELLING Georges Duffa 1684, chemin des Clos 83136-Gar´eoult, France

Abstract We describe the gas-material physical and chemical reactions appearing at the surface of a re-entry vehicle and in the core when this material contains a resin which pyrolyse under the influence of thermal front propagation. The heterogeneous reactions are described for a carbon-based material and different levels of description of the phenomenon are given. The first part is devoted to the physical and chemical interaction of carbon with different gases. In a second part it is shown that these effects conduct to three-dimensional surfaces due to heterogeneity of material. Methods to treat this phenomenon using an equivalent flat surface are described. The third part is devoted to analyse the macroscopic local problem in order to understand the consequences of previous phenomena in engineering calculations.

1

Nomenclature B′ c˜ D CH CM EA f h kf , kb

= = = = = = = = =

m ˙ M N Ns p q˙ T u R ρ ω˙ Ψ θ

= = = = = = = = = = = = =

reduced mass flux, elemental mass fraction, diffusion coefficient, m2 · s−1 Stanton number for heat flux, Stanton number for mass transfer, activation energy, J · mol−1 volume fraction, mass enthalpy, J · kg−1 gaseous reaction constant, (m3 · mol−1 )α · s−1 or heterogeneous reaction constant, m−2 · s−1 · P aβ mass flux, kg · m−2 · s−1 molecular mass, kg · mol−1 Avogadro number, N = 6.02214179 × 1023 mol−1 surface density of sites, m−2 pressure, P a heat flux, W · m−2 temperature, K velocity, m · s−1 universal gas constant, R = 8.314472 J · mol−1 · K −1 density, kg · m−3 mass flux by unit surface, kg · m−2 · s−1 or unit volume, kg · m−3 · s−1 surface fraction, fraction of free or occupied sites

The numerical values are CODATA values [1]. α and β are related to stœchiometry.

1

Introduction: description of some carbon-materials used

We are interested here by carbon-carbon or carbon-resin materials which an important class used for re-entry TPS (dual-layer C-C, TWCP, PICA, etc.) and for which a quasi-complete modelling of ablation can be made. To a certain extent materials using cork fall into this category even if the material itself is not totally known (and then its chemical decomposition). We put then aside materials using fibres or additives of SiO2 , W2 O3 , CaO, Al2 O3 , B2 O3 (Avcoat, Aleastrasil, etc.) where a liquid phase is present during ablation, leading to important modeling difficulties not totally solved today. The material can have a periodic geometry like TWCP (Fig. 1) which is a woven material or have an random aspect like PICA (Figure 2). Both use a rayon fibre (Fig. 3) used for its low conductivity value [5] and a phenolic resin. After pyrolysis this resin give a glassy carbon poorly organized [6, 7] contrary to the fibre (Fig. 4). The heterogeneous reactions between gas and material take place on these graphene surfaces whose reactivity depends on orientation: only the extremities of the structure can react. It should be noted that the flat surface of material generally considered for modelling has no

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physical existence and can be defined as an asymptotic equivalent problem. The definition of the scheme conducting to this equivalent surface has to be given.

2

Heterogeneous reactions at carbon surface

The carbon surface react with species present near its surface like O2 , N2 but also CO2 , H2 O or C2 H2 coming from the atmosphere or products of decomposition of resin. At higher temperatures it sublimes giving mainly linear carbon radicals C1 , C2 , C3 , etc.

2.1 Chemical heterogeneous reactions 2.1.1

Mechanisms

The different parts of graphene structure have very different reactivities. The plane surface is theoretically non-reactive. In fact the numerous defects in the structure (Fig. 5) or the presence of adatoms are sufficient to promote a reactive region (generally a hole) where the extremities of structure can be accessed. These edges of graphene are of two types (Fig. 6) each of them having a different reactivity. These locations are the regions of material where reactions takes place and are denoted Site and can be treated as a special species in modelling. The material is characterized by the number of sites per unit area Ns and the proportion of them unoccupied θ or occupied by the species A (often reduced to a single atom) θA . There are various types of interaction which we cite the more important: • adsorption (fixation of A on a site), for example Site + O → Site − O Site + O2 → 2 Site − O this second example corresponding to a complex mechanism of dissociative adsorption of dioxygen. • desorption (the inverse phenomenon) Site − O → Site + O • Eley-Rideal mechanism Site − O + O ⇋ Site + O2 or Site − O + CO ⇋ Site + CO2 • Langmuir-Hinshelwood mechanism involving two neighboring sites 2 Site − O ⇋ 2 Site + O2

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One can also note that the molecules or atoms attached to the surface can move on it by surface diffusion. This mechanism will not intervene directly on the quantities selected for describe the phenomena (Ns and θ). But it is obvious that the phenomenon of diffusion, by spreading occupied sites on the surface will alter the effectiveness of the Langmuir-Hinshelwood mechanism. We assume also that a molecule (or atom) adsorbed on a site does not change its environment, especially the properties of neighboring sites. This last item is not really true in the case of ablation or deposition on carbon, as demonstrated by molecular dynamics. The reactions presented above correspond to mass conservation of solid. Ablative reactions can be modelised in the same way. For example for oxygen: Site − O (+ Cs ) → Site + CO and Site − O + O (+ Cs ) → Site + CO Cs is a solid carbon. The above mechanism written for oxygen can also be used for nitrogen even if the reactivities are very different in both cases. Note that the carbon-nitrogen are much less known, in particular dissociative adsorption have not be described. 2.1.2

Reaction kinetics

The reaction kinetics are derived from microscopic analysis showing the rate of efficient phenomena per unit area and time. We analyze each of these mechanisms. • Adsorption In the case of collision of a gas molecule with the wall we have d d (Ns θ) = − (Ns θA ) = ω˙ A = −ΦA γA θ dt dt

(1)

This equation simply says that the number of molecules ”stuck” per unit area and time is proportional to 1. ΦA , the number of collisions per unit area and time of the molecule A with the wall, so, assuming a Maxwellian velocity distribution ΦA =

N pA 1

(2πMA RT ) 2

(2)

Maxwellian distribution is a good approximation of the exact distribution (ChapmanEnskog distribution [10, 9]) when the flow is continuous. We neglect the effects of Knudsen layer. 2. the probability θ as the receiver site is free. In fact, the sticking probability is superior to θ. This is likely related to the potential attractiveness of sites occupied. This is particularly true at low temperature: this effect will be ignored here. 4

3. the effectiveness of the collision γA (or ”sticking” probability), constant or given by a term of Arrhenius type representing the effect of the potential barrier between the average gas molecule and the surface of the material [11]   EA γA = α f (T )exp − (3) RT In this case the reaction coefficient is defined by d (Ns θ) = −kf pA θ dt

(4)

d (Ns θ) = −kf pA θ dt

(5)

and equation (1) becomes

In this latest writing, the reaction coefficient kf becomes a data, for example of experimental origin. • Desorption In the case of desorption (Eyring law, giving the probability that the thermal agitation bring sufficient energy to break the bond [12]) d RT d (Ns θ) = − (Ns θA ) = Ns γA θA = kf θA dt dt hN

(6)

where the term RT hN γA represents the frequency of thermal desorption of a site and Ns θA the number of sites occupied by the element that interests us. The reaction coefficient is kf = Ns

RT γA hN

(7)

γA being described by an Arrhenius law. • Eley-Rideal mechanism For this mechanism involving an atom or molecule B in the gas phase to give a gaseous product AB we have d d 1d (Ns θ) = − (Ns θA ) = − (Ns θB ) = Ns ΦSAB θA θB −θ 2 γb ΦAB = kf θA θB −kb θ 2 pAB 2 dt dt dt (8) with  S  kf = Ns ΦAB (9)  AB kb = γbpΦAB where ΦSAB is the formation frequency of a molecule from atoms (or molecules) attached to two neighboring sites. 5

• Langmuir-Hinshelwood mechanism The molecule B present in gaseous medium and interacting with an occupied site forms a molecule AB carried away (and the reverse phenomenon, for example the dissociative adsorption of O2 ) d d (Ns θ) = − (Ns θA ) = θA γf ΦB − θγb ΦAB = kf θA pB − kb θpAB dt dt

(10)

γf is the reaction probability for direct interaction. This type of interaction must overcome a potential barrier: this term is of Arrhenius type. For a backward reaction the probability γb also includes a potential barrier. The reaction constants are therefore of the form  γf ΦB   kf = pB   k = b

• Other mechanisms

(11)

γb ΦAB pAB

Note the possibility for a molecule to bind to two neighboring sites [11]. This type of mechanism is advanced when an intermediate of the reaction can explain experimental observations [14]. It may perhaps explain schemes referring to sites of different nature [13]. In addition, surface diffusion, which tends to distribute evenly occupied sites, can alter rates reactions by the Langmuir-Hinshelwood mechanism by decreasing the probability that two neighboring sites are occupied. Note that: • the classical catalytic recombination of oxygen on a non-ablative surface is a particular case of what precede, but is often simplified supposing negligible dissociation of dioxygen. This simplification can give important errors on a discontinuity of material [21]. • it is not possible in general to reduce the mechanism of oxydation or nitridation to a simple law m ˙ = f (pO2 ) or m ˙ = f (pN2 ) as cited in literature. These laws are generally correlations coming from a limited experimental domain. 2.1.3

Modelling the ablative system

A very general way to write this type of heterogeneous reaction is given in [20]. We consider the site as an ”element” and the free or occupied site as a ”species”. The nR reactions (including nA ablation reactions) are written in the case where nS different types of sites are present For k = 1, nr ne X i=1

′ νik Ai

+

ns X

µ′jk Sitej

⇋

ne X i=1

j=1

6

′′ νik Ai

+

ns X j=1

µ′′jk Sitej

(12)

Conservation of sites

and

N ω˙ j d (NS θj ) = dt Mj

(13)

 P  j θj

(14)

 P

Production rates are given by

= 1

˙j = 0 jω

ω˙ j = −Mj

nr X

Ωjk τk

(15)

k=1

where Ω is the stœchiometric matrix. N τk = kfk

ne Y

ν′

pi ik

i=1

ns Y

µ′

θl lk − kbk

ne Y

ν ′′

pi ik

i=1

l=1

ns Y

µ′′

θl lk

(16)

l=1

The ablation mass rate is given by the equation nA X τj m ˙ C = MC Mj

(17)

j=1

We can obviously neglect the gain or loss of mass due to catalytic reactions. 2.1.4

Database

Reactions kinetics for oxygen are relatively well known and can be find in [22, 23]. The situation is very different with nitruration for which opposite values were published. Park [24, 25] give high values resulting from experiments and making this phenomenon important unlike Suzuki [26] whose experiments make this phenomenon secondary: the measured sticking coefficient is 100 times less. Note that the equilibrium constants of these reactions are not independent one from another and of gaseous reaction constants. Unlike of existing databases for gases the coherence of data has to be verified. A general procedure is proposed in [18]. The problem of knowledge of site surface density is treated in Section 3.

2.2 Sublimation 2.2.1

Mechanism

The phase change can be described by a Langmuir-Knudsen mechanism [15] m ˙ i = αi



Mi 2πRT

7

1

2

(peqi − pi )

(18)

The kinetic approach of sublimation above is based on the idea that one can replace the solid  1 2 Mi with a gas at the equilibrium vapour pressure peqi . Indeed, 2πRT pi is the incident mass flux. m ˙ i = 0 at thermodynamic equilibrium implies that the flux emitted by the wall is equal to 1  2 Mi peqi . αi is significantly less than unity. 2πRT This approach seems indicate that the inverse process (deposition of carbon) can be described by the same mechanism. It is probably not realistic and one has to take care with this phenomenon which can appear in calculations (in a region of relaxation of pressure for example). Note that this description does not involve the notion of site even if it is possible to introduce this kind of mechanism [18]. In fact the atomic description of sublimation is unknown. 2.2.2

Database Table 1: Carbon sublimation: ”sticking” coefficients [19] Species C C2 C3

αi (2700 K) (basal plane) 0.24 0.50 0.023

αi (2450 K)

αi (2500 K)

0.37 0.34 0.08

0.14 − 0.23 0.26 − 0.38 0.03 − 0.04

The αi are given by various experimental sources, independent of the temperature, but different for each species. Only one author gives values for various faces of a monocrystalline graphite (table 1, last column, in the order: base plane, edge). It is noticed that measurements were made at relatively low temperatures compared to the applications need which requires temperatures going over 4000 K. Indirect measurements allow to confirm these values at the temperatures of use (Section 3).

2.3 Carbon deposition Carbon deposition is present in all materials where a pyrolysis take place (resins, cork). The mass deposition is important only in some cases for example Avcoat material Fig. 7). The possible mechanisms were studied in combustion and CVI (Carbon Vapour Infiltration) problems which conclude to the main role played by acetylene. Due to the impossibility of direct sticking on a non-participative surface, a complex mechanism was given, named HACA (Hydrogen Abstraction, Carbon Addition). This mechanism is presented in Fig. 8. With some simplification [18] it is possible to demonstrate that this mechanism can be represented by a simple law giving a mass flux deposition proportional to the partial pressure of acetylen m ˙ = 5.0 × 10−22

8

Ns pC2 H2 T

(19)

It is to note that molecular dynamics simulations authorize direct sticking of acetylen with a modification of the graphene structure during and after reaction (dissipation of the excess of energy by phonons).

2.4 Some results In the case of graphite in air the behaviour of system is simple: • at low temperatures the sites are almost entirely occupied by oxygen atom until thermal desorption (Fig. 9). At higher temperatures the sites are free and act as a catalyst. • the carbon react with oxygen to give mainly CO. At low temperature this species react with oxygen to give CO2 more stable. Due to elevated speed of this reaction the system is rapidly driven by diffusion in boundary layer. The global result for re-entry ablation is then not very sensitive to the kinetics employed for this reaction. • at higher temperatures CN is formed by heterogeneous reactions and then other nitrides Cn Nm (mainly C2 N, C2 N2 , C4 N2 ) by gaseous reactions. • for temperatures higher than 3000 K carbon sublimes to give linear Cn , mainly C3 . Note that the hypothesis of chemical equilibrium generally used to present this kind of results is not verified for temperatures lower than 2000 − 2500 K. In the case of carbon phenolics not only the same species as above are present at surface but also species coming from the hydrogen present in the system, mainly H2 , H, HCN , Cn H radicals and a great number of hydrocarbons Cn Hm (Fig. 9) among which C2 H2 plays a specific role (Section 2.3). At lower temperatures heavier hydrocarbons (for example CH4 , C3 H3 , C2 H6 ) and water are present and the medium is out of chemical equilibrium. For calculation it is also necessary to add species created by gaseous reactions in hight temperature regions of boundary layer (in particular ions) leaving to heavy models [28].

3

Real and equivalent surface

The concept of a plane interface between material and flow do not correspond to what one observe at mesoscopic (submillimetric) level, even for pure graphite. In laminar flow the surface is rough with typical ogival geometries of fibres or yarns (Fig. 10). In fact the notion of a flat surface of material comes from an asymptotic description always made but never justified. It is evident that the fibre and the carbonaceous residue of pyrolysis have different structures then different densities and reactivities. This explain the roughness setup but we need to go further: • what kind of geometry is created? • is a stady-state stable geometry existing? • what is the influence of this geometry on ablation? 9

We put aside the influence of this roughness on convective problems (turbulence) and restrain ourselves to the region near the wall where diffusion-reaction problems takes place. The Figure 10 explain the formation of an ogival geometry for a fibre or a yarn perpendicular to surface for a pure reactive problem (no diffusion limitation): • the pyrolytic carbon is more reactive than those of fibre and recedes more rapidly. • the fibre is ablated on the top and faces until the formation of a cone. k

If the reactions are first-order, the angle of the cone is given by sin θ = ρρRf kRf where f is the fibre and R the coke. Note that the roughness is independent of the flow and created within a recession of the same order than its height, then very rapidly. When the diffusion plays a role, the analytical solution (stable steady-state geometry) of the problem is an ogival geometry with a characteristic radius r = ρD β , where β characterize the
21 M reaction (Fig. 11). In the case of sublimation β = α 2πRT p [27]. In the more general case of inclined fibres the same result is obtained by numerical calculation (Fig. 11) [29]. This value not vary much in the region of the trajectory where roughness can play a role on convection. In this sense it is possible to speak about ”the roughness of a material”. Note that this geometry can be used to infer values of sticking coefficients at temperatures much higher that those obtained in experiments for which databases where issued (≃ 2500 K). We are now able to discuss the equivalent properties of equivalent smooth surface provided some approximations [30]: • the diffusion flux lines are vertical. This hypothesis is well verified on vertical fibres or yarns. • first-order irreversible reactions with Reynolds number sufficiently low. Then the equivalent reactivity if function of reactivities of fibres and coke only and independent of the flow. If these approximations are verified then the equivalent reactivity of the system fibre+coke is comprised between • the algebraic mean keq = Ψkf +(1−Ψ)kR where Ψ is the projected part of surface occupied by the fibre or the yarn when the system is dominated by reactions. −1  when the system is dominated by diffusion. • the harmonic mean keq = kΨf + 1−Ψ kR In this last case, given the much higher value of coke reactivity we can conclude that it drives the ablation. One has to be careful with this observation: this is true only for given material temperature and does not authorize any conclusion on what happens when reactivity of the weak part of material is modified. To conclude it is necessary to look at global energetic mechanisms.

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4

Global mechanisms

4.1 Approximations In preceding sections we were interested in what happens at nanoscopic or mesoscopic scales. Now we look at the global response of the system constituted by the material and its environment for a given location on the re-entry vehicle at a given time on the trajectory. We will use standard approximations with • the hypothesis of chemical equilibrium for gases typically obtain for Tw > 2000 − 2500 K. • the mass exchange in the boundary layer defined by the factor ρe ue CM . • the relationship between reduced surface mass ablation B ′ = flow Bg′

m ˙g ρe ue CM

m ˙ ρe ue CH ,

the reduced mass py-

and the elemental mass fractions c˜i (w=wall, s=solid, g=pyrolysis = rolysis gas, e= boundary layer top) (B ′ + Bg′ + 1) c˜kw = B ′ c˜ks + Bg′ c˜kg + c˜ke

(20)

On this base the ablation system can be described by a law f (T, p, B ′ , Bg′ , ρe ue CM ) = 0 (Fig. 12)[18]. For practical purpose, this law is simplified by supposing chemical equilibrium between solid and gas, making disappear the term ρe ue CM . Note that this is an inconsistent hypothesis: in this case the ablation is zero. This can see as a asymptotic case: m ˙ → 0 and ρe ue CM → 0 simultaneously making B ′ constant (same thing for Bg′ ). Looking at Figure 12 one can see that this is a rough approximation making an error up to 200 K for a given m ˙ in the sublimation regime. What is the effect of this error on the global result? An answer is obtained observing that steady-state ablation hypothesis give a good approximation of the problem, at least in the most severe part of re-entry. In this situation it is possible to write an energy conservation equation [18]. ρe ue CH (ha − hw ) + q˙R −

ρp [hp (Tw ) − hv (T0 )] − ǫσTw4 = 0 ρv

(21)

where ha is the adiabatic enthalpy, hp and hv are pyrolyzed and virgin enthalpies of material. T0 is the room temperature and then the term hv (T0 ) is almost negligible. The first term is the convective term corrected for composition and blowing, the second term represent incident radiative flux, the third ablation, pyrolysis and conduction, the last term correspond to the energy re-radiated. We suppose a small effect of blowing on radiation [31]. Using a linearized approximation of blowing correction CH = CH0 − ηB ′ we have   ρp (hp − hv (T0 )) = ρe ue CH0 (ha − hw ) + q˙R (22) m ˙ ρe ue η(ha − hw ) + ρv

4.2 Consequences Using this approach, some general characters of ablation problems can be given:

11

1. This kind of analysis allows us to understand the importance of each term in equation 21, depending mainly to Tw [18, 32]: • the re-radiation represent 60 to 80% of incident flux. • all reactions including pyrolysis represent 15 to 50%. • the conduction is limited to 10 to 20%. 2. The equation 22 shows that in the region where the variation of m ˙ versus Tw is rapid, m ˙ is almost proportional to the total incident heat flux. This fact is verified by exact calculations. The consequence is that for a given material the total ablated mass is almost only dependant of its composition. An example is given on Fig. 5 were composition of a PICA-like material was varied and tested on the stagnation point of Stardust up to drogue ejection. In a large domain of density (here 165 to 660 kg · m−3 ) obtained by varying fibre volume fraction fF and resin fraction fR the ablated mass varies only of 10%. 3. The important effect of ρe ue CM on ablation law (Fig. 12) induces limited errors or ablation speed. A shift of 200 K on temperature give an error of about 5% on ablation, experimentally undetectable. The same kind of uncertainty is associated to high temperature thermodynamic properties of carbon. Note that this shift is also difficult to detect experimentally due to uncertainty of emissivity and the difficult calibration of measurements at these high temperatures. 4. This mechanism explains also that an important variation of reactivity of material (the problem treated in Section 3) has no great influence on ablation: the difference of reactivity is compensated by a difference of temperature, the ablation being almost unchanged.

5

Conclusion

From this presentation one can give some conclusions on actual problems, focusing on light PICAlike materials: • a survey and upgrade of existing data of carbon high temperature thermodynamic properties will be welcome and is possible with existing data. • the problem of conflicting data on nitrogen-carbon reaction kinetics has to be solved. • noting that the re-radiation plays a major effect in ablation balance the probable gap between gas and material temperatures [33, 34] must be studied, this effect lowering the re-radiated flux. • the semi-transparency of light materials has the same consequence when the radiative mean T free path is comparable to thermal characteristic length ||∇T || [35]. This effect can induces a reduction of radiation up to 20%.

12

Figure 1: Tissue (TWCP). Figure 2: PICA morphology [3, 4].

Figure 4: Structure of fibre [2].

Figure 3: Rayon fibre.

13

Figure 5: Nanometric structure of carbon fibre [8]. Orange regions correspond to imperfect graphene structures.

Figure 6: Reactive sites of graphene.

Avcoat

PICA

Figure 7: Densities of pyrolysed materials [16, 17].

14

Figure 8: HACA mechanism

Species

Site coverage

Figure 9: Species present in ablation and site coverage (carbon-resin, thermodynamic equilibrium)

15

Ablated fiber (or yarn) formation dominated by reaction Ablated fiber (experiment) Figure 10: Creation of roughness (1)

Ablated fiber (calculation)

Ablated fiber (or yarn) formation dominated by reaction-diffusion mechanism

Figure 11: Creation of roughness (2)

16

Reduced mass flux (ρe ue CM in kg · m−2 · s−1 )

Surface mass ablated (Stardust trajectory)

Figure 12: Ablation of a carbon phenolic (steady state). fF and fR are volume fractions of fibre and resin.

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