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Acta Astronautica

Acta Astronautica 100 (2014) 40–46

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First-principle study of atomic oxygen and nitrogenadsorption on (1 1 1) β-cristobalite as a modelof thermal protection coverage

A.A. Buchachenko a, A.A. Kroupnov b, V.L. Kovalev b,c,n

a Department of Chemistry, M.V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow 119991, Russian Federationb Institute of Mechanics, M.V. Lomonosov Moscow State University, Michurinskii Prosp. 1, Moscow 119192, Russian Federationc Department of Mechanics and Mathematics, M.V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow 119991,Russian Federation

a r t i c l e i n f o

Article history:Received 18 February 2014Received in revised form10 March 2014Accepted 16 March 2014Available online 24 March 2014

Keywords:Thermal protection coatingHeterogeneous reactionsSilica polymorphsCristobaliteAtomic oxygenAtomic nitrogen

x.doi.org/10.1016/j.actaastro.2014.03.01165/& 2014 IAA. Published by Elsevier Ltd. A

esponding author at: Institute of MechaniState University, Michurinskii Prosp. 1, Mos

ion. Tel.: þ7 495 939 37 54.ail address: valerykovalev@yandex.ru (V.L. K

a b s t r a c t

Atomic oxygen and nitrogen adsorption on the Si site of the (1 1 1) face of idealβ-cristobalite is studied within the cluster model using the density functional approach.The M06 and B3LYP functionals are found to be the most appropriate for studying atomicadsorption on silica. The calculations show significant difference in adsorption propertieswith respect to previously studied (1 0 0) β-cristobalite face, but do not reveal remarkableeffect of the surface relaxation.

& 2014 IAA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Proposed Space missions and reusable Space vehicleprograms require, among other developments, design ofefficient thermal protection systems. The heat released to avehicle's surface by heterogeneous chemical reactions inthe shock wave at Earth's atmosphere re-entry is practi-cally the only component of the grand thermal balanceamenable to minimization by a smart material engineering[1–4]. The lack of real flight experiments and limitations ofEarth-based measurements have placed the first-principlemultiscale modeling at the forefront of this development.

ll rights reserved.

cs, M.V. Lomonosovcow 119192, Russian

ovalev).

Indeed, last decade has revealed continuous growth ofattention to the solid theoretical backgrounds in Chemis-try- and Material Science-related aspects of the problem.

Silica-based ceramic materials, tested during and inrelation to the first reusable spacecraft programs,obviously provide the useful playground for assessing thetheoretical approaches. Recent density functional theory(DFT) calculations on the atomic oxygen and nitrogenrecombination on the β-cristobalite and quartz surfaces[5–12] have led to essential revision of the mechanismsand rate parameters derived not only by purely empiricalmodels (see, e.g., Refs. [13–16]), but also by semiempiri-cal approaches [17]. In particular, adsorption energy ofatomic oxygen was proven to be much larger thanaccepted before; as far as adsorption is the key step ofany heterogeneous catalytic mechanism, this has dramaticconsequences for the whole reaction chain leading to

A.A. Buchachenko et al. / Acta Astronautica 100 (2014) 40–46 41

recombination. Sayós and co-workers explored the solid-state periodic slab model of the ideal (1 0 0) β-cristobalitecleavages, an approach that allowed them to map all thestationary points of the interaction potential energy sur-face (PES) and find several minima, representing distinctadsorption sites, and saddles, representing reaction ordiffusion barriers. Polymorph models explored by Ruti-gliano et al. [6,8] mimic the electronic density at thesurface and, owing to their simplicity, permit one to usemore elaborated quantum chemistry methods. Despite animpressive agreement achieved at the intersection of twostreams, many issues remain for further exploration.

First, it is not clear what surface model (if any) betterrepresents the real thermal protection coating at re-entrycondition—amorphous (e.g., reaction-cured glass testedwithin the Space Shuttle program [3,18]), corrugated andlikely subjected to thermal and chemical degradation.β-Cristobalite modification is assumed to be a good model[5] mostly because of the similarity of its properties toamorphous silica, high thermal stability and existence ofthe Earth-based high-temperature recombination experi-ments [19,20]. However, uncertainty exists in the relativestability of various polymorphic modifications of β-cristo-balite, see, e.g., Refs. [21,22]. Furthermore, silica poly-morphs are known to have conchoidal fracture: theirsurfaces correspond to erratic combination of differentfaces rather than to a single ideal one [23,24]. Differentfaces and their cleavages should interact with the shock-wave particles distinctly. Second, both periodic and poly-morph models are not well adapted to reproduce surfacerelaxation upon interactions with ambient species. Third,though DFT is an obvious method of choice for large-scalemodeling, no test has been made so far on the perfor-mance of different functionals for the problem at hand.

In this paper, we consider alternative model for atomicoxygen and nitrogen adsorption on silica. We representthe particular adsorption site (Si of the Si-terminated(1 1 1) β-cristobalite face) by the saturated cluster, whichpermits us to model both the electronic density distribu-tion at the adsorption site of interest and relaxation effectsdeemed to be plausible in a real material. Preliminarytesting of the density functionals is performed to elucidatethe uncertainties for adsorption site geometry and energycalculations. The results of the DFT study and simplestatistical modeling of adsorption equilibrium reveal thatthe Si sites on the (1 1 1) and (1 0 0) β-cristobalite faces

Fig. 1. Side (x, z) view of the Si10O12H15 cluster. Numbers refer to the (x, y) planewhite balls. (For interpretation of the references to color in this figure legend, t

have remarkably different adsorption properties withrespect to both O and N atoms. Our model does notdemonstrate important effect of the cluster relaxation.

2. Cluster model

We consider the Si-terminated (1 1 1) cleavage ofβ-cristobalite assuming its fcc Fd3m bulk structure [25],in contrast to Refs. [5–10] that addressed (1 0 0) face of thesimilar or the same crystal. It should be noted thatappearance of the (1 1 1) face in the conchoidal cristoba-lite surface was estimated as 58%, whereas that of(1 0 0)—as 11% only [26]. Essential difference between twoSi-terminated faces is that the Si site has one unsaturatedbond at the (1 1 1) face instead of two at the (1 0 0) one.Consideration of this site thus provides a valuable additionin identification of the range of adsorption energies typicalto a real ceramics. Furthermore, we used the clustermodel, which, like a polymorph model, allows one tomimic the surface electron density distribution locally atthe designated site: previous works on atomic oxygenadsorption in much less ambiguous case of the Al site atthe (0 0 0 1) face of hexagonal close packed α-Al2O3 crystalrevealed impressive agreement between the periodic andcluster models [27–29]. In addition, cluster model permitsone easy assessment of the relaxation effects upon surfaceformation and adsorption, as well as testing of various DFTschemes.

Being viewed from the top, the Si-terminated (1 1 1)cleavage of β-cristobalite consists of equilateral trianglesmerged in hexagons, with the outer Si atoms located invertices and the second-layer Si atoms—in the center ofeach triangle. To describe adsorption over the outer Siatom, we retained three Si–O–Si bridges to the second Silayer, three bridges to the third layer and two bottombridges per atom. The resulting structure shown in Fig. 1consists of 10 Si, 12 O and 15H atoms saturating thedangling Si–O bonds. It has C3v point symmetry group(according to the Fd3m bulk structure, all the Si–O bondsare equivalent) and bears one unpaired electron (total spinSc¼1/2).

All calculations were performed by the DFT methodwith the functionals and basis set described below. For anisolated Si10O12H15 cluster, two configurations were con-sidered, the “bulk” one, optimized by keeping equal all theSi–O bond lengths, and “surface” one, where four inner

s of Si (dark, red) and O (gray) atoms. Hydrogen atoms are shown as smallhe reader is referred to the web version of this article.)

A.A. Buchachenko et al. / Acta Astronautica 100 (2014) 40–4642

(Si, O, Si, O) (x, y) planes were frozen, 2nd Si plane (seeFig. 1 for designations) and top Si atom were allowed toshift along vertical z direction, and the oxygen atomsbelonging to the 1st plane were able to move consistentlyalong the z direction and the C3v vertical symmetry planes.Fixed parameters were taken from the “bulk” cluster. Thelengths and directions of the artificial Si–H bonds wereoptimized ones for “bulk” cluster and kept frozen in allsubsequent calculations. Adsorbed O or N atom wasapproached the top Si atom along z direction. Optimizationrestrictions on the cluster relaxation upon adsorption werethe same as for the “surface” cluster.

3. Results

3.1. Choice of the density functionals

Test on the performance of different gradient-correctedand hybrid density functionals were undertaken by con-sidering the diatomic molecules relevant to the problem,namely, O2, N2 and NO, as the products formed in theprospective recombination studies, and SiO, as roughlyrepresenting interactions in the material. Preliminaryscreening indicated that above the 6-31Gn level of thebasis set the results exhibit much stronger dependence onthe choice of the functional rather than on further expan-sion of the basis. It also allowed us to reduce the numberof the functionals to six, including similar to those used inthe previous periodic and polymorph calculations [5–12].Table 1 presents the equilibrium distances Re, dissociationenergies De and harmonic frequencies ωe as the relativepercentage deviations Δ from the reference values [30]obtained with the functionals B1LYP [31–33], B3LYP[31,32,34], M06 [35], MPW1LYP [36,37] and MPW1PW91[36–38]. Accuracy of the functionals per molecule isroughly represented by Δ3, the sum of three absolute Δvalues, whereas overall performance is given by the sum of

Table 1Performance of the density functionals for diatomic molecules. Reference valvibrational frequency ωe (cm�1) were taken from Ref. [30]. For calculated valux¼re, De and ωe. Δ3 represents the sum of the absolute Δ values for three co“Sum” entry.

Molecule Const. Reference value Relative perc

B1LYP

O2(3Σ�g ) re 1.20752 0.3

De 5.214 �1.8ωe 1580.2 6.4Δ3 8.5

NO(2Π) re 1.15077 0.5De 6.615 �4.2ωe 1904.1 5.6Δ3 10.3

N2(1Σþg ) re 1.09768 0.6

De 9.905 �5.1ωe 2358.6 5.0Δ3 10.7

SiO(1Σþg ) re 1.5097 0.7

De 8.337 �8.8ωe 1242 1.4Δ3 10.9

Sum Δ3 40.4

Δ3 values for four molecules. It is evident that B3LYPprovides the best accuracy on average and for thenitrogen-containing test molecules, while calculationswith M06 better reproduce the properties of the O2 andSiO molecules.

3.2. Free cluster

Geometry of the “bulk” cluster is given by the singleparameter of the Si–O bond length d. Our calculations gave1.632 (B1LYP), 1.633 (B3LYP), 1.623 (M06), 1.631(MPW1LYP) and 1.628 Å (MPW1PW91), in significantexcess of the experimental value d¼1.55 Å [25,39]. Over-estimation of the cell volume by the periodic DFT calcula-tions is notorious for different silica polymorphs [40,41].For instance, Ref. [5] gives 1.614 and 1.615 Å for PW1 andPBE calculations, respectively. From the comparison of thePW1-based calculations one can infer that the clustermodel further stretches the Si–O bound by 0.02 Å withrespect to the periodic bulk calculations. M06 functionalgives the shortest bond length and may be considered asthe most adequate, especially taking into account its bestperformance for the Si–O bond length in the diatomicmolecule, see Table 1.

Infrared (IR) spectrum calculated with the M06 func-tional consists of five transitions (those related to termi-nating H atoms excluded; for other functionals frequenciesvary within 20 cm�1). The first three at 1223, 1192,994 cm�1 correspond well to the broad band in the IRemission spectra centered at 1100 cm�1 and, for α-cristo-balite, resolved in 1200, 1110 and 1070 cm�1 triplet [42].This band, present also in the IR and Raman spectra[43,44], corresponds to Si–O–Si antisymmetric stretchingmodes. The Si–O–Si symmetric stretch vibrations givethe band at 800 cm�1, while the O–Si–O bending—at490 cm�1 [42]. M06 calculation gives 727 and 555 cm�1,

ues of equilibrium distance re (Å), binding energy De (eV) and harmonices the relative percentage error Δ(x)¼(xcalc�xref)�100/xref are presented,nstants. The sums of the Δ3 values for all four molecules are given in the

entage error

B3LYP M06 mPW1LYP mPW1PW91

0.6 �0.1 0.3 �0.43.8 1.1 �0.2 1.25.0 9.1 6.5 9.49.3 10.3 7.0 11.00.7 0.4 0.5 0.1

�0.02 �3.9 �2.9 �3.34.6 6.8 5.7 7.45.3 11.1 9.1 10.80.8 0.7 0.5 0.4

�2.4 �6.0 �4.2 �5.54.2 4.6 5.1 5.87.4 11.3 9.9 11.70.9 0.4 0.7 0.5

�6.1 �3.3 �7.8 �9.10.4 3.1 1.4 2.37.4 6.8 9.9 7.9

29.4 39.5 35.9 41.4

Table 2Adsorption energies Ea(O), Ea(N) and Si–O(N) distances zO, zN for atomic oxygen and nitrogen adsorbed on the Si-sites calculated within different models ofsilica surfaces. For literature data, “mixed” Ea values are presented; “adiabatic” ones are given, whenever available, in parentheses.

Calculation Ea(O), eV zO, Å Ea(N), eV zN, Å

Si10O12H15 cluster representing the Si-terminated (1 1 1) β-cristobalite surfaceB1LYP 4.963 1.678 3.416 1.795B3LYP “bulk” 5.029 1.673 3.456 1.792B3LYP 5.051 1.677 3.472 1.795M06 “bulk” 5.183 1.668 3.481 1.779M06 5.208 1.672 3.500 1.782mPW1LYP 5.035 1.677 3.477 1.793mPW1PW91 4.948 1.675 3.389 1.792Periodic model for T1 site on the Si-terminated (1 0 0) β-cristobalite surface [5]PW91 S¼0 or 1/2 5.97 1.527 2.88 1.649PW91 S¼1 or 3/2 3.18 1.526 0.04 1.650Polymorph model for the Si-terminated (1 0 0) β-cristobalite surface [6,8]PBE0 mixed (S¼0) 5.57 (8.24) 1.52 – –

B3LYP mixed (S¼1/2) – – 2.79 (5.62) 1.65Polymorph model for the Si-terminated (1 0 0) β-quartz surface [12]PBE0 mixed (S¼0) 5.58 (8.08) 1.52 – –

A.A. Buchachenko et al. / Acta Astronautica 100 (2014) 40–46 43

respectively. Very reasonable agreement with experimentsupports the adequacy of the cluster model.

Relaxation of the “bulk” cluster to the “surface” oneleads to outward z-displacements of the top Si atom andthe 1st (O) plane by ca. 5 and 3%, respectively, andnegligible relaxation of the 2nd (Si) plane. Optimizationrestrictions likely exaggerate the stiffness of the cluster:for instance, periodic slab model of the (1 0 0) faceprovided 9.3% outward relaxation of the surface plane[5]. However, cutting one Si–O bond by the (1 1 1) planemay indeed lead to smaller deformation than cutting thetwo Si–O bonds along the (1 0 0) face.

3.3. Oxygen and nitrogen attachment

Interaction of the doublet Si10O12H15 cluster (spinSc¼1/2) with the triplet atomic oxygen (SO¼1) or quartetnitrogen (SN¼3/2) gives the states with the doublet andquartet or triplet and quintet multiplicities (total spinS¼1/2 and 3/2 or S¼1 and 2), respectively. The calcula-tions revealed that for both atoms two states correlate tothe same asymptotic limit zO or zN-1 and the state withthe lower spin has deep minimum. Atom-cluster interac-tions with the higher spin were found to be almostrepulsive. In contrast, previous calculations on the (1 0 0)polymorphs [6,8], which bear two excess electrons, gavedifferent asymptotic limits for two multiplicities. Thestates with lowest multiplicity (singlet for adsorbed oxy-gen and doublet for nitrogen) have deep minima, but go toexcited asymptotic limits. Adsorption energy can bedefined as “mixed”, relative to the lowest limit, or as“adiabatic”, relative to the asymptote for the proper multi-plicity. In accord with our findings, the states with highestmultiplicity did not exhibit prominent adsorption minima.In the periodic calculations, deep minima were found forboth multiplicities (magnetizations of the surface) [5], butsimilar crossover from singlet to triplet character withincreasing zO was mentioned for oxygen [7]. It is thereforereasonable to consider adsorption energies reportedtherein as “mixed”.

Table 2 summarizes our results and the data fromprevious calculations. Oxygen adsorption energies varyfrom 4.95 to 5.20 eV depending on the choice of thefunctional, with the maximum value obtained in the mosttrusted M06 calculations. The range is narrower for nitro-gen, 3.39–3.48 eV. The distances between adsorbed and Siatoms are reproduced within 0.01 Å. Our model predictsoxygen adsorption energies by ca. 3 eV lower than the“adiabatic” ones in the (1 0 0) polymorph model, whereasadsorbate-surface distances—by 0.15 Å longer. The numberof unsaturated bonds at the adsorption site is the mainfactor, as confirmed by the polymorph model of the Si-terminated (1 0 0) β-quartz surface [12], see Table 2.Indeed, the difference in the surface geometry does notstrongly affect the adsorption parameters of the sites withtwo cropped Si–O bonds. On the other hand, our adsorp-tion energies are only 0.4–0.8 eV lower for oxygen andeven larger for nitrogen than the “mixed” ones on the(1 0 0) face [5,6,8].

The frequencies of the adsorbed O and N atoms wereestimated, at the B3LYP level, as 182, 239, 869 cm�1 and177, 178, 795 cm�1, respectively; other functionals gavevery similar results. These values are in line with thosereported for (1 0 0) periodic calculations [5], 120, 216, 948(O) and 191, 242, 805 (N), though smaller splittingbetween two lowest frequencies that correspond to themotions parallel to the surface is worthy of noting for themore symmetric (1 1 1) Si-site.

In order to elucidate the effects of cluster relaxation,selected calculations were carried out for adsorption onthe frozen bulk cluster. The representative B3LYP and M06results marked as “bulk” are also shown in Table 2.Attachment of oxygen (nitrogen) atom leads to significantcluster deformation: with the B3LYP functional, the top Siatom is displaced inward by 6.1% (5.9%), whereas the 1st(O) plane—outward by 2.7% (3.4%). The dependence ofadsorption properties on cluster relaxation is, however,weaker than that produced by the alteration of the func-tionals. The difference in the vibrational frequenciesof adsorbed atoms does not exceed 10 cm�1. This analysis

0.0005 0.0010 0.0015 0.002020

30

40

50

60

70

80

90O

ln(K

) (cm

3 mol

-1)

1/T (K)

(111) M06 (111) B3LYP (111) B3LYP "bulk" (100) periodic

N

Fig. 2. Arrhenius plot of the equilibrium constant for oxygen and nitrogen adsorption on the (1 1 1) and (1 0 0) β-cristobalite faces. Calcu-lations for the latter used the periodic results from Ref. [5].

500 1000 1500 2000 2500

2x1011

3x1011

4x1011

5x1011

6x1011

O

N

(111) M06 (111) B3LYP (111) B3LYP "bulk" (100) periodic

k a (c

m3 m

ol-1

s-1)

T (K)

Fig. 3. Oxygen and nitrogen adsorption rate coefficients for the (1 1 1) and (1 0 0) β-cristobalite faces. Calculations for the latter used the periodic resultsfrom Ref. [5].

A.A. Buchachenko et al. / Acta Astronautica 100 (2014) 40–4644

justifies the omission of the relaxation effects in theprevious periodic and polymorph models.

In Ref. [5], Sayós and co-workers quoted their unpublishedperiodic results on adsorption energies for the Si-terminated(1 1 1) β-cristobalite cleavage, namely 5.45 (2.72) eV for oxy-gen on the singlet (triplet) PES and 3.08 (1.00) eV for nitrogenon the doublet (quartet) PES. They confirm, qualitatively, thatthe (1 1 1) adsorption of oxygen is weaker, while that ofnitrogen is stronger, than the (1 0 0) one. Empirical force-fieldmodel was reported to estimate oxygen adsorption energy onthe same Si site of the (1 1 1) face as 2.1 eV [45].

3.4. Equilibrium adsorption

First-principle studies of atomic and molecular adsorp-tion can be used for fitting global analytical PESs, which, inturn, make it possible to evaluate elementary rate coeffi-cients using classical or semiclassical methods. It has beenproven [6–10,12] that this approach is capable to describethe kinetics of dominant heterogeneous chemical pro-cesses at re-entry conditions. For the purpose of qualita-tive comparison between the (1 1 1) and (1 0 0) faces ofβ-cristobalite, we adopted here much simpler statisticalapproaches that assume thermal equilibrium.

At temperature T, common to gas phase and surface,the equilibrium adsorption constant K, equal to the ratio of

adsorption to desorption rate coefficients ka/kd, isexpressed as [46]

KðTÞ ¼ hg expðEa=kBTÞ ð2πmkBTÞ�1=2ðqCa=qCÞ ðNA=S0Þ; ð1Þ

where m is the mass of adsorbed atom, qC and qCa are the(vibrational) partition functions of the cluster and clusterwith atom adsorbed, respectively, g is the statistical weightwhich accounts for spin (for oxygen atom, also orbital)degeneracy, S0 is the number of adsorption sites per unitsurface square, and common notations are used for con-stants. Making the simplification that internal vibrationalmotions of the cluster remain intact upon adsorption (anatural extension of the unrelaxed surface approximationthat allowed us to handle the results of Ref. [5]), it ispossible to express qCa/qC ratio using three frequencies ofadsorbed atom. The numbers of Si sites S0 were estimatedfrom the geometry of ideal Fd3m faces as 2.9�1014 cm�2

(1 1 1) and 3.6�1014 cm�2 (1 0 0).Adsorption rate coefficient can be evaluated separately

using the transition state theory for non-activated adsorp-tion, see, e.g., Ref. [47]:

kaðTÞ ¼ h2kBTð2πmkBTÞ�3=2qS; ð2Þ

where qS is the partition function associated to the vibra-tional motion of adsorbed atom in the surface plane.

A.A. Buchachenko et al. / Acta Astronautica 100 (2014) 40–46 45

Selected results for equilibrium constant in the 500–2500 K temperature range are presented, as Arrheniusplots, in Fig. 2. Adsorption equilibrium is dominated bythe exponential factor containing Ea (1), being thereforeshifted towards oxygen desorption for (1 1 1) face withrespect to (1 0 0) face. Nitrogen adsorption shows oppositetrend being more competitive with oxygen one for the(1 1 1) β-cristobalite face. Variations with the densityfunctional and suppression of surface relaxation (“bulk”cluster model) appear to be much less pronounced.

Adsorption rate coefficient (see Fig. 3) is determined bythe statistical factor qS, whose sensitivity to surface struc-ture is comparable with the variations in surface modeland DFT scheme.

4. Summary and conclusions

Atomic oxygen and nitrogen adsorption on the Si site ofthe ideal β-cristobalite (1 1 1) face is studied within thecluster model using the density functional approach. Themodel provides reasonable characterization of the struc-ture and vibrational spectrum of the bulk material. Assess-ment of the density functionals indicates that M06 isoptimal for the oxygen case, whereas B3LYP functionaldescribes oxygen and nitrogen adsorption on the equalfooting.

Oxygen adsorption on the Si sites is found to be weakerfor (1 1 1) face than for (1 0 0) one [5,6,8], where theunpaired electron from the second cropped bond furtherenhances the atom-surface bonding. In the case of nitro-gen, the reverse relation indicates repulsive contributionof the excess electron density.

Cluster model points out that surface relaxation effectsare not very significant, justifying the previous theoreticaldata and simplifying future studies on heterogeneousreactions on silica.

From the thermodynamics prospective, the main dif-ference between (1 1 1) and (1 0 0) faces corresponds tothe variation of adsorption energies, which manifests itselfin the shifts of adsorption equilibrium, of opposite signsfor oxygen and nitrogen. It should have remarkable con-sequences on the modeling of heterogeneous reactions ofthe shock-wave particles at ceramic coatings of reusableSpace vehicle at re-entry conditions.

Acknowledgments

This work was supported by the Russian Basic ResearchFoundation (projects 11-01-00280 and 14-01-00310).

References

[1] R. Goulard, On catalytic recombination rates in hypersonic stagna-tion heat transfer, J. Jet Propuls 28 (1958) 737–745.

[2] M. Capitelli (Ed.), Molecular Physics and Hypersonic Flows, NATO ASISeries C, vol. 482, Kluwer, Dordrecht, 1996.

[3] V.L. Kovalev, Heterogeneous Processes in Aerothermodynamics,Fizmatlit, Moscow, 2002.

[4] V.L. Kovalev, O.N. Suslov, G.A. Tirsky, Phenomenological theory forheterogeneous recombination of partially dissociated air on high-temperature surfaces, in: M. Capitelli (Ed.), Molecular Physics and

Hypersonic Flows, NATO ASI Series C, vol. 482, Kluwer, Dordrecht,1996, pp. 193–202.

[5] C. Arasa, P. Gamallo, R. Sayós, Adsorption of atomic oxygen andnitrogen at β-cristobalite (1 0 0): a density functional theory study, J.Phys. Chem. B 109 (2005) 14954–14964.

[6] M. Rutigliano, A. Pieretti, M. Cacciatore, N. Sanna, V. Barone, N atomsrecombination on a silica surface: a global theoretical approach,Surf. Sci. 600 (2006) 4239–4246.

[7] C. Arasa, H.F. Busnengo, A. Salin, R. Sayós, Classical dynamics studyof atomic oxygen sticking on the β-cristobalite (1 0 0) surface, Surf.Sci 602 (2008) 975–985.

[8] M. Rutigliano, C. Zazza, N. Sanna, A. Pieretti, G. Mancini, V. Barone,M. Cacciatore, Oxygen adsorption on β-cristobalite polymorph: Abinitio modeling and semiclassical time-dependent dynamics, J. Phys.Chem. A 113 (2009) 15366–15375.

[9] C. Arasa, V. Morón, H.F. Busnengo, R. Sayós, Eley–Rideal reactiondynamics between O atoms on β-cristobalite (100) surface: a newinterpolated potential energy surface and classical trajectory study,Surf. Sci 603 (2009) 2742–2751.

[10] V. Morón, P. Gamallo, L. Martin-Gondre, C. Crespos, P. Larregaray,R. Sayós, Recombination and chemical energy accommodationcoefficients from chemical dynamics simulations: O/O2 mixturesreacting over a β-cristobalite (0 0 1) surface, Phys. Chem. Chem.Phys. 13 (2011) 17494–17504.

[11] I. Armenise, M. Rutigliano, M. Cacciatore, M. Capitelli, Hypersonicboundary layers: oxygen recombination on SiO2 starting from abinitio coefficients, J. Thermophys. Heat Transf 25 (2011) 627–632.

[12] C. Zazza, M. Rutigliano, N. Sanna, V. Barone, M. Cacciatore, Oxygenadsorption on β-quartz model surfaces: some insights from densityfunctional theory calculations and semiclassical time-dependentdynamics, J. Phys. Chem. A 116 (2012) 1975–1983.

[13] O. Deutschmann, U. Riedel, J. Warnatz, Modeling of nitrogen andoxygen recombination on partial catalytic surfaces, J. Heat Transf.(Trans. ASME) 117 (1995) 495–501.

[14] F. Nasuti, M. Barbato, C. Bruno, Material-dependent catalytic recom-bination modeling for hypersonic flows, J. Thermophys. Heat Transf10 (1996) 131–136.

[15] A. Daiß, H.-H. Frűhauf, E.W. Messerschmid, Modeling of catalyticreactions on silica surfaces with consideration of slip effects, J.Thermophys. Heat Transf 11 (1997) 346–352.

[16] T. Kurotaki, Catalytic model on SiO-based surface and application toreal trajectory, J. Spacecraft Rockets 38 (2001) 798–800.

[17] M. Cacciatore, M. Rutigliano, G.D. Billing, Eley–Rideal and Langmuir–Hinshelwood recombination coefficients for oxygen on silica sur-faces, J. Thermophys. Heat Transf 13 (1999) 195–203.

[18] V.L. Kovalev, A.F. Kolesnikov, Experimental and theoretical simula-tion of heterogeneous catalysis in aerothermochemistry (a review),Fluid Dyn 40 (2005) 669–693.

[19] M. Balat-Pichelin, J.M. Badie, R. Berjoan, P. Boubert, Recombinationcoefficient of atomic oxygen on ceramic materials under earth re-entry conditions by optical emission spectroscopy, Chem. Phys 291(2003) 181–194.

[20] L. Bedra, M. Rutigliano, M. Cacciatore, M. Balat-Pichelin, Atomicoxygen recombination on quartz at high temperature: experimentsand molecular dynamics simulation, Langmuir 22 (2006) 7208–7216.

[21] S. Liu, S.H. Garofalini, R.D. King-Smith, D. Vanderbilt, First-principlesstudies on structural properties of β-cristobalite, Phys. Rev. Lett. 70(1993) 2750–2753.

[22] S. Coh, D. Vanderbilt, Structural stability and lattice dynamics of SiO2

cristobalite, Phys. Rev. B: Condens. Matter 78 (054117) (2008) 1–14.[23] C. Frondel, Dana's system of mineralogy, seventh ed. Silica Minerals,

vol. III, Wiley, New York, 1963.[24] A.P. Legrand (Ed.), The Surface Properties of Silica, Wiley, New York,

1997.[25] R.W.G. Wykoff, Crystal Structures, vol. 1, Wiley, New York, 1963.[26] V.V. Murashov, E. Demchuk, Surface sites and unrelaxed surface

energies of tetrahedral silica polymorphs and silicate, Surf. Sci 595(2005) 6–19.

[27] V.L. Kovalev, A.A. Kroupnov, M. Yu. Pogosbekyan, L.P. Sukhanov,Simulation of oxygen atom heterogeneous recombination on Al2O3

from ab initio approach, Prog. Flight Phys 3 (2012) 351–364.[28] A.A. Buchachenko, V.L. Kovalev, A.A. Kroupnov, Closed model of

oxygen recombination on the Al2O3 surface, Russ. J. Phys. Chem. B 32(2013) 86–94.

[29] B. Hinnemann, E.A. Carter, Adsorption of Al, O, Hf, Y, Pt, and S atomson α-Al2O3(0 0 0 1), J. Phys. Chem. C 111 (2007) 7105–7126.

[30] K.-P. Huber, G. Herzberg, Molecular Spectra and Molecular Structure.IV. Constants of Diatomic Molecules, Van Nostrand Reinhold,New York, 1979.

A.A. Buchachenko et al. / Acta Astronautica 100 (2014) 40–4646

[31] A.D. Becke, Density-functional exchange-energy approximation withcorrect asymptotic behavior, Phys. Rev. A: At. Mol. Opt. Phys. 38(1988) 3098–3100.

[32] C. Lee, W. Yang, R.G. Parr, Development of the Colle–Salvetticorrelation-energy formula into a functional of the electron density,Phys. Rev. B: Condens. Matter 37 (1988) 785–789.

[33] C. Adamo, V. Barone, Toward reliable adiabatic connection modelsfree from adjustable parameters, Chem. Phys. Lett 274 (1997)242–250.

[34] P.J. Stephens, F.J. Devlin, C.F. Chabalowski, M.J. Frisch, Ab initiocalculation of vibrational absorption and circular dichroism spectrausing density functional force fields, J. Phys. Chem. 98 (1994)11623–11627.

[35] Y. Zhao, D.G. Truhlar, The M06 suite of density functionals for maingroup thermochemistry, thermochemical kinetics, noncovalentinteractions, excited states, and transition elements: two newfunctionals and systematic testing of four M06-class functionalsand 12 other functionals, Theor. Chem. Acc 120 (2006) 215–241.

[36] J.P. Perdew, Unified theory of exchange and correlation beyond thelocal density approximation, in: in: P. Ziesche, H. Eschrig (Eds.),Electronic Structure of Solids, 91, Akademie Verlag, Berlin, 1991,pp. 11–20.

[37] C. Adamo, V. Barone, Exchange functionals with improved long-range behavior and adiabatic connection methods without adjus-table parameters: the mPW and mPW1PW models, J. Chem. Phys108 (1998) 664–675.

[38] B.J. Lynch, Y. Zhao, D.G. Truhlar, Effectiveness of diffuse basisfunctions for calculating relative energies by density functionaltheory, J. Phys. Chem. A 107 (2003) 1384–1388.

[39] D.A. Keen, M.T. Dove, Local structures of amorphous and crystallinephases of silica, SiO2, by neutron total scattering, J. Phys.: Condens.Matter 11 (1999) 9263–9273.

[40] Th. Demuth, Y. Jeanvoine, J. Hafner, J.G. Ángyán, Polymorphism insilica studied in the local density and generalized-gradient approx-imations, J. Phys.: Condens. Matter 11 (1999) 3833–3874.

[41] M. Catti, B. Civalleri, P. Ugliengo, Structure and energetics of SiO2

polymorphs by quantum-mechanical and semiclassical approaches,J. Phys. Chem. B 104 (2000) 7259–7265.

[42] K.S. Finnie, J.G. Thompson, R.L. Withers, Phase transitions in cristo-balite and related structures studied by variable temperature infra-red emission spectroscopy, J. Phys. Chem. Solids 55 (1994) 23–29.

[43] G.L. Hua, T.R. Welberry, R.L. Withers, Lattice dynamics of α- andβ-cristobalite, SiO2, J. Phys. Chem. Solids 50 (1989) 207–213.

[44] L.P. Huang, J. Kieffer, Molecular dynamics study of cristobalite silicausing a charge transfer three-body potential: phase transformationand structural disorder, J. Chem. Phys 118 (2003) 1487–1498.

[45] S. Di Benedetto, C. Bruno, A novel kinetic model for chemical energyaccommodation on silica surface, in: Presented at Fourth EuropeanConference for Aerospace Sciences (2011).

[46] S. Glasstone, K.H. Laidler, H. Eyring, The Theory of Rate Processes,McGraw-Hill, New York, 1941.

[47] I. Chorkendorff, J.W. Niemantsverdriet, Concepts of Modern Catalysisand Kinetics, Wiley-VCH, Weinheim, 2007.

Alexei A. Buchachenko was born in 1965. He received the Candidate ofScience in Physics and Mathematics (Ph.D.) degree in 1993 and Doctor ofScience in Physics and Mathematics (Hab.) degree in 2003 at theDepartment of Chemistry, M.V. Lomonosov Moscow State University.He is the professor of the Department of Chemistry. His main researchinterests are intermolecular interactions, collision dynamics and mole-cular spectroscopy.

Alexandr A. Kroupnov was born in 1960. He received the Candidate ofScience in Physics and Mathematics (Ph.D.) degree in 1995 at the Instituteof Mechanics, M.V. Lomonosov Moscow State University. He is the leadingresearcher of the Laboratory of Nanomechanics of this Institute. Hisresearch interests include hypersound aerodynamics, heat and masstransport, catalytic processes, quantum chemistry.

Professor Valery L. Kovalev was born in 1952. He received the Candidateof Science in Physics and Mathematics (Ph.D.) degree in 1985 and Doctorof Science in Physics and Mathematics (Hab.) degree in 1997 at theDepartment of Mechanics and Mathematics, M.V. Lomonosov MoscowState University. He is the professor of the Department of Mechanics andMathematics and head of the Multiscale Simulation Laboratory. He alsoleads the Laboratory of Nanomechanics at the Institute of Mechanics,M.V. Lomonosov Moscow State University. His main research interests arethe mechanics of nanosystems, fluid dynamics, hypersound aerody-namics, heat and mass transfer, catalytic activity, multiscale simulationof complex mechanical processes.