DTIC ADA558696: Modeling Oxidation Kinetics of SiC-Containing Refractory Diborides

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by  Defense Technical Information Center

J.Am.Ceram.Soc.,95[1]338–349(2012)
DOI:10.1111/j.1551-2916.2011.04927.x
Journal ©2011TheAmericanCeramicSociety
Modeling Oxidation Kinetics of SiC-Containing Refractory Diborides
T. A. Parthasarathy,‡,§,† R. A. Rapp,¶ M. Opeka,k and M. K. Cinibulk‡
‡AirForce Research Laboratory,Materials andManufacturing Directorate, Wright-PattersonAFB, Ohio 45433-7817
§UES, Inc., Dayton,Ohio 45432
¶TheOhio State University,Columbus, Ohio 43235
kNaval SurfaceWarfareCenter, Carderock,Maryland 20817
Experimental data on the oxidation kinetics of SiC-containing cate glass that fills the pores of the porous oxide scale and
diborides of Zr and Hf in the temperature regime of 1473– often also results in a glassy external layer, as seen from
2273 K are interpreted using a mechanistic model. The model Fig 1(a) (courtesy Prof. E Opila, University of Virgina). In a
encompasses counter-current gas diffusion in the internal SiC furnace under static air, a continuous external glassy layer is
depleted zone, oxygen permeation through borosilicate glass formed, but at high temperatures, it begins to flow due to a
channels in the oxide scale, and boundary layer evaporation at decrease in viscosity giving rise to some scatter in experimen-
the surface. The model uses available viscosity, thermodynamic tal data. The underlying porous oxide is almost always filled
and kinetic data for boria, silica, and borosilicate glasses, and with the borosilicate glass. Underneath this oxide layer,
a logarithmic mean approximation for compositional varia- some investigations have reported a depleted region where
tions. The internal depletion region of SiC is modeled with SiC is absent but the diboride is still intact as, for example,
CO/CO counter diffusion as the oxygen transport mechanism. seen in Fig. 1(a). The depleted region is absent at low tem-
2
Data reported for pure SiC in air/oxygen, for ZrB containing peratures, but has been reported in samples exposed to high-
2
varying volume fractions of SiC, and for SiC–HfB ultra-high temperature air. However, there is an inconsistency in the
2
temperature ceramics (UHTCs) by differentinvestigations were reported data with some studies not finding any depleted
compared with quantitative predictions of the model. The layers under the same conditions where others have observed
model is found to provide good correspondence with labora- them.7,10,14,20,21 In some work, a partially depleted region
tory-furnace-based experimental data for weight gain, scale consisting of SiC and a Si–O–C phase, is reported. The high-
thicknesses, and depletion layer thicknesses. Experimental data est temperatures are achieved under arc-jet conditions, where
obtained from arc-jet tests at high enthalpies are found to fall typically very thick oxide layers with a very thin or no exter-
well outsidethe modelpredictions, whereas lowerenthalpydata nalglassylayerarereportedforveryshorttimes.Adepletion
were closer to model predictions, suggesting a transition in zone has also been observed under these conditions. Some
mechanism inthe arc-jetenvironment. works have suggested that convection currents can be signifi-
cant during oxidation.17–19 To obtain a quantitative model
thatinterprets these datawasthe objective ofthis work.
I. Introduction In an initial study, we reported on a model for the oxida-
tion of monolithic diborides of Zr and Hf in the temperature
REFRACTORY diborides, especially ZrB2 and HfB2, with region of 1073–2673 K.22–24 In the present work, we add the
SiC additions are being studied with great interest
complexities arising from the presence of SiC and its oxida-
because of their high thermal conductivity, high melting
tion products, silica and CO. We retain all the essential ele-
point, and moderate resistance to environmental degradation
ments of the prior model, including volumetric effects from
in air at very high temperatures.1–3 This combination of
phase change of the oxide, and transitions in mechanistic
properties of these two-phase composites makes them prom- regimes as temperature is increased.24 As in the prior model,
ising for use in hypersonic vehicles as leading edge compo-
the refractory oxide in the scale is assumed to be imperme-
nents, which are subject to high heat fluxes.1 Several
able to oxygen due to the known low electronic conduc-
additions and compositional refinements are being tried to tivity.22 The effects of the presence of SiC are the formation
enhance their oxidation resistance, but SiC-containing ZrB
2 of a borosilicate glass instead of a boria glass with all its
or HfB are the most studied, with sufficient data to enable a
2 consequences, the formation and transport of gaseous CO
modeling-basedanalysisandunderstandingoftheiroxidation
and SiO,and the evaporation ofSiO and SiO at the surface.
behavior.4–19 2
The properties of silica and boria are known but modeling
A brief summary of experimental observations is as fol-
their variation with composition needs interpolation. The
lows.4–19 The oxidation behavior is dominated by parabolic
vapor pressures of B O and SiO are known as a function
kinetics despite formation of a porous oxide of the refractory 2 3 2
of temperature, and that of SiO is known as a function of
metal. This protective behavior is attributed to the borosili-
temperatureand oxygenpartial pressure.
The model is able to interpret most of the experimental
data reported on ultra-high temperature ceramics (UHTCs).
Weight gain, oxide thickness, external glass thickness, and
N.Jacobson—contributingeditor internal depletion layer thickness for both SiC–ZrB and SiC
2
–HfB are found to be in reasonable agreement with experi-
2
mental data, but with some exceptions mostly related to
arc-jet test data. The equations that constitute the model are
ManuscriptNo.29879.ReceivedJune15,2011;approvedOctober03,2011.
ThisworkwassupportedbyUSAFContract#FA8650-10-D-5226whichincluded presented first; this is followed by a section where the predic-
funding from US Air Force Office of Scientific Research (AFOSR), monitored by tions are compared with the available experimental data.
Dr.AliSayir.
†Authortowhomcorrespondenceshouldbeaddressed.e-mail:triplicane.parthasar- The final section discusses the merits and limitations of the
[email protected]
338
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January 2012 Oxidation Modeling of SiC-Containing Refractory Diborides 339
(a)
(b)
l l l
12 23 3a
1 2 3 a
5
MeB2 2O2 MeO2 B2O3 MeO O2 Boundary
2 Si Layer
1fs MeB2 B diffusion SiO2(g)
SiO(g)
BO(g)
2 3
O (dissolved)
fs SiC SiO/CO,CO2 B2O32SiO2 CO
(assumed
i1 i2 i3 a Notratelimi(cid:415)ng)
SiC 2CO2 SiO 3CO SiO 12O2 SiO2 SBi2OO23((ll)) SBi2OO23((gg))
2CO O2 2CO2 SiO2(l) SiO(g)
(c)
1 2 3i a
MeO
2
MeB Evapora(cid:415)onlimitedby
2
Diffusionthrough
Porouschannels
SiO (g)
O (diss.) 2
SiC SiO/CO,CO B O2 SiO SiO(g)
2 2 3 2 B O (g)
2 3
i1 i2 i3i a
Fig.1. (a) SEM image of the microstructure of oxidation scale formed on a ZrB–SiC sample (Courtesy: E Opila, Univ. Virginia) (b, c)
2
Schematic sketches of the oxidation products and morphology assumed in the model. At lower temperatures (b) external glassy scale forms,
whereasathighertemperaturesorhighambientflow(c),theglassyscalerecedesinwardduetoevaporativelossofSiO andBO.
2 2 3
model, including possible reasons for discrepancies and porous channels of glass, eventually reaching the interface i2,
suggestionsfor future work. where it reacts with the diboride. A portion of the oxygen
flux, assumed to be proportional to areal (= volume) fraction
of SiC, is transported within the depleted region (void of
II. The Model
solid but filled with gases) to oxidize SiC through a medium
(1) Model Framework and Key Assumptions of a gas mixture of CO and CO , similar to the model by
2
The morphology of the oxidation scale assumed in the model Holcomb and St. Pierre for HfC oxidation.26 The gaseous
wasderived from published cross-sectional microstructures of reaction product, CO, of the SiC oxidation is assumed to
oxidized UHTCs.4,5,8,10,13,21,25 The key elements of the model either diffuse or bubble through the glassy scale (as observed
are shown in Fig. 1. Figure 1(a) shows a sample micro- in ref. 17–19) and is assumed not to be rate limiting, consis-
structure of the scale formed on a ZrB –SiC material. The tent with prior assumptions for oxidation of SiC.27 Fig-
2
illustration, in Figs. 1(b) and (c), details the assumed mor- ure 1(b) depicts the expected scenario at higher temperatures
phology of the oxidation products, representing steady-state (likely above 2000 K) where evaporation of B O and SiO
2 3 2
conditions. The substrate is a composite of SiC and MeB from the external surface is sufficiently high that an external
2
(Me = Zr, Hf) with f being the volume fraction of SiC. The glassy layer cannot be supported and the glassy layer recedes
s
oxidation product, viz. the scale, consists of disconnected (in inwards. In all regimes, the diffusivity of gases in a multi-
cross-section) MeO grains with a continuous porous (void component gas mixture and the effect of Knudsen diffusion
2
of solid) region that is filled with liquid borosilicate. [when anunfilledpore waspresentasin Fig. 1(b)]were mod-
Throughout the manuscript, pore refers to a region void of eledas detailed in prior work.22
MeO but filled with glass, except at very high temperatures The volume fraction of SiC in the substrate is f, and the
2 s
when glass evaporates leaving pores behind [Fig. 1(c)]. The volumefractionofporosity(filledwithglass)withintheMeO
2
MeO grains are taken to be impermeable to oxygen, as regionisf .Asinpriorwork,theporefraction,f ,(filledwith
2 p p
explained in our prior work.22 Molecular oxygen from the glass)istakentochangeasthetemperaturecrossesthemono-
ambient dissolves in borosilicate as molecular O , diffuses clinic–tetragonal phase transformation temperature, T .
2 trans
across the external glassy phase, and permeates through the The parameters f and f are related to the volume fraction,
s p
340 Journal of the American Ceramic Society—Parthasarathy et al. Vol. 95, No. 1
f ,ofMeO intheglass + MeO region(2–3),andfraction
oMfegOl2ass,f ,(bo2rosilicate)inthescal2easfollows. K ¼ aSiO2(cid:2)i2 ;
g SiO P ðP Þ1=2
SiO(cid:2)i2 O2(cid:2)i2
ffpg¼¼ffs1fþorfpTð1\(cid:2)TtfrsaÞns andf2 forT[Ttrans (1) KSiC ¼PPSiSOiC(cid:2)ði1PðCPOC2O(cid:2)(cid:2)i1i1Þ2Þ3; (4)
f ¼ð1(cid:2)f Þð1(cid:2)fÞ¼1(cid:2)f ðP Þ2
MeO2 p s g K ¼ CO2(cid:2)i2
CO2 ðP Þ2P
As shown in Fig. 1, (1 (cid:2) f) is the volume fraction of CO(cid:2)i2 O2(cid:2)i2
s
MeB and (1 (cid:2) f ) is the volume fraction of MeO within
the M2eO + glassplayer (region 2–3). Thus, when no2 SiC is Atsteadystate,the fluxes ofthe gaseous specieswithin the
2 depleted zone (between i1 and i2) must be decided and be
present (monolithic MeB ), f = f . In the presence of SiC,
2 g p consistent with the equilibrium constants of Eqs. (4), and the
the glassvolume fraction isincreased byf.
s oxidation rate is limited by incoming oxygen flux at i2,
The B O activity, a , of the borosilicate scale at the
iMnetBer2m–Ms2eoOf32thientreercfeascsei,oBdn2eOrn3a(cid:2)otit2eedofbtyhethMe seuBbscprhipatsei,2,dRisMedB2e,riavnedd JTOh2u(cid:2)s3,2ftoirmtehsetdheepalerteeadfzroanctei,own,efos,btoafinS:iC at the interface i2.
2 dt
that of the SiC phase, dRSiC. Accounting for their phase frac-
tSiioCn,sananddastshuemirinmgoalnaridvedoatllummiexst,urVe,MweBe2,obVtSaiiCn:of MeB2 and jJCO(cid:2)12j¼32jJCO2(cid:2)21j; jJSiO(cid:2)12j¼12jJCO2(cid:2)21j;
2
(cid:1) (cid:3) jJ j¼ fjJ j
ð1(cid:2)fÞ dR SiO(cid:2)12 3 s O2(cid:2)32
s MeB2 (5)
aB2O3(cid:2)i2¼ð1(cid:2)fsÞ(cid:1)VdMReMBe2B2(cid:3)þdtfs (cid:1)dRSiC(cid:3); (2) jJCO2(cid:2)21j¼fsDDCROT2(cid:2)1210P5PCO2(cid:2)(cid:2)i2l(cid:2)P12PCO2(cid:2)i1;
aSiO2(cid:2)i2¼V1M(cid:2)eB2aB2O3(cid:2)di2t VSiC dt jJCO(cid:2)12j¼fs CROT(cid:2)12105 CO(cid:2)i1l12 CO(cid:2)i2
In these expressions, J is the flux in moles per unit area
Here, the activity is taken to be directly proportional to
per unit time, D the diffusivity, R the universal gas
molar ratio, assuming an ideal behavior. This assumption is
constant, T the absolute temperature, and l is the length
fairly consistent with the work of Boike et al., who reported 12
of the depleted zone. The factor 105 refers to conversion
only a slight positive deviation from ideal behavior and the
from atm to Pascal. The first subscript used in all the
experimental work of Kawamoto et al., who found no phase
separationin this B O –SiO system.28,29 quantities refers to the species. The second subscript used
2 3 2 with partial pressures refers to the interface, while that
The equilibrium constant for the oxidation of diboride is
used with diffusivity and flux refers to the regions between
related to the boria activity and oxygen partial pressure at
interfaces.
interface i2and isgivenby:
There are seven unknowns that describe region 1–2, of
5 which the oxygen partial pressure at i2 is given by Eq. (3).
Ati2:MeB2þ2O2¼MeO2þB2O3; The remaining six unknowns, the partial pressures of the
a a (3) three species (CO, CO , and SiO) at the two interfaces i1 and
K ¼ MeO2 B2O3(cid:2)i2 ; 2
Me a ðP Þ5=2 i2, are obtained by solving the Eqs. (4) and (5). Thus, we
MeB2 O2(cid:2)i2 obtain the following expressions for these six unknowns.
They are presented in algorithmic sequence, wherein each
where K refers to equilibrium constant, a the activity, and P
variable can be calculated from a knowledge of all preceding
the partial pressure, whereas subscripts refer to species and
variables.
the interface i2shown inFig. 1.
(cid:1) (cid:3)
a a 2=5 a
P ¼ MeO2 B2O3(cid:2)i2 ; P ¼ SpiO2ffiffi(cid:2)ffiffiiffi2ffiffiffiffiffiffiffiffi;
(2) Internal Depletion Region (1–2) O2(cid:2)i2 aMeB2KMe SiO(cid:2)i2 KSiO PO2(cid:2)i2
Due to the low oxygen partial pressure at interface i2, the
2RTjJ jl
only way SiC can oxidize at interface i1 behind i2, is if PSiO(cid:2)i1¼3(cid:3)105OD2(cid:2)32 12 þPSiO(cid:2)i2
oxygen was transported across the void by CO/CO counter SiO(cid:2)12
2
diffusion. Likewise, a gaseous SiO(g) from oxidation of
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
SiC, transports Si to sustain the glassy oxidation product at
P ðD D K P Þ
i2. The oxygen dissolved in the glass at the interface i2 CO(cid:2)i1 CO(cid:2)12 CO2(cid:2)12 COr2 ffiffiffiOffiffi2ffiffi(cid:2)ffiffiiffi2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rreeagciotsn w(1it–h2)CtOo(go)xitdoizeforSmiCCaOt2(ign)tewrfhacicehi1d.iffTushees SaicOro(gss) (cid:2)ðPCO(cid:2)i1Þ3=2DCO(cid:2)12DCO2(cid:2)12 KCOK2PSiO(cid:2)i1
SiC
formed at interface i1 diffuses back to interface i2 where it
(cid:2)2D D P
is oxidized by the dissolved oxygen in the glass. This model CO(cid:2)12 SiO(cid:2)12 pSiOffiffiffi(cid:2)ffiffiiffi1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
yields the following reactions with associated equilibrium (cid:2)3D D K P P
CO2(cid:2)12 SiO(cid:2)12 CO2 O2(cid:2)i2 SiO(cid:2)i1
constants.
þ2D D P
CO(cid:2)12 SiO(cid:2)12 pSiOffiffiffi(cid:2)ffiffiiffi2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ3D D K P P ¼0
Ati2:SiOðgÞ þ2O2ðdissolvedÞ¼SiO2; CO2(cid:2)1ð2P SiO(cid:2)1Þ23=2pffiPCffiffiOffiffiffi2ffiffiffiffiffiOffiffiffiffi2(cid:2)i2 SiO(cid:2)i2
2COðgÞþO2ðdissolvedÞ¼2CO2ðgÞ PCO2(cid:2)i1¼ CO(cid:2)i1pffiKffiffiffiffiffiffiffiffi SiO(cid:2)i1
SiC
3
Combining:SiOðgÞþ2COðgÞþ2O2ðdissolvedÞ PCO(cid:2)i2¼DCO(cid:2)12PCO(cid:2)i1(cid:2)3DSiOD(cid:2)12PSiO(cid:2)i1þ3DSiO(cid:2)12PSiO(cid:2)i2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CO(cid:2)12
¼SiO þ2CO ðgÞ
2 2 P ¼P K P
Ati1:SiCþ2CO ðgÞ¼SiOðgÞþ3COðgÞ CO2(cid:2)i2 CO(cid:2)i2 CO2 O2(cid:2)i2
2
(6)
January 2012 Oxidation Modeling of SiC-Containing Refractory Diborides 341
One of them, P , requires numerical solution of the temperature and oxygen partial pressures of interest (see for
CO-i1
implicit equation. The only unknowns in the above equation example, ref. 7).
set are the oxygen flux, jJ j, in the region 2–3 (Fig. 1),
O2(cid:2)32 (cid:1) (cid:3)
oanbdtaitnheedlefnrgotmh osofltvhinegdetphleeteedquzaotnioen,sl12th.aTtheapflpulyx jtJoO2r(cid:2)e3g2ijoins dl3a ¼ dl23 fMeO2 (cid:2)(cid:5)(cid:5)J (cid:5)(cid:5) V þðjJ j
2–3, discussedbelow. dt(cid:5) dt(cid:5)VM(cid:5)eO2 B(cid:5)2O3(cid:2)vap B2O3 SiO(cid:2)12
(cid:2)(cid:5)J (cid:5)(cid:2)(cid:5)J (cid:5)ÞV (cid:2)dl23f
SiO2(cid:2)vap SiO(cid:2)vap SiO2 dt g
(T3h)e eMqueaOti2on+sBth2Oat3–gSoiOve2rnGltahses Roxeyggioenn(fl2u–x3)in region 2–3, (cid:5)(cid:5)Jspecies(cid:2)vap(cid:5)(cid:5)¼DRspeTcies105Pspdecies(cid:2)vap;dbdry (9)
betweeninterfaces i2 andi3,are givenbelow. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(cid:6) (cid:7) bdry
¼3 lspecimen gflluid 1=6ðD Þ1=3
2 v q species
fluid fluid
P (cid:2)P
jJ j¼P O2(cid:2)i3 O2(cid:2)i2f
O2(cid:2)32 O2(cid:2)B2O3(cid:2)SiO2 l g
23 In the above equation set, the subscript ‘vap’ refers to
jJO2(cid:2)a3j¼PO2(cid:2)B2O3(cid:2)SiO2ð3aÞPO2(cid:2)al(cid:2)3aPO2(cid:2)i3 evvraepfeorrattoiovnis,clospseictiym,edneinsstithye,laenndgtvheloofcitthyeosfptehceimaemnb,igen,tρ,fluanidd.
Since;jJ j¼jJ j The last equation gives the boundary layer (thickness d)-
O2(cid:2)a3 O2(cid:2)32 dependentevaporation of species.30
l P P þf l P P
P ¼23 O2(cid:2)B2O3(cid:2)SiO2ð3aÞ O2(cid:2)a g3a O2(cid:2)B2O3(cid:2)SiO2 O2(cid:2)i2 The viscosity of the glassy layer could limit the maximum
O2(cid:2)i3 l23PO2(cid:2)B2O3(cid:2)SiO2ð3aÞþfgl3aPO2(cid:2)B2O3(cid:2)SiO2 thickness, l3a-max, of the external layer that can be supported
(7) under gravitational forces. As in the previous work,24,31 from
the theory of falling liquid films,31 under laminar flow, this
limiting thickness isgivenby:
Here, Π refers to the permeability (rate of transport in
moles per unit area per unit time per unit partial pressure " #1=3
tghraatdiienntr)e.gTiohne2p–e3r,maesarbeigliitoynin3-areigsiolinke3ly-atoisbediffaebreonrita-fdroefim- l3a(cid:2)max¼ 3ðMB2O3(cid:2)SgiOq22CB2O3(cid:2)SiOsi2nlsðp/ecÞÞgB2O3(cid:2)SiO2
cient region resulting from faster evaporation of boria than B2O3(cid:2)SiO2 (10)
dl 1
silica. C ¼ 3a
The unknowns that remain are the three lengths, l12, l23, B2O3(cid:2)SiO2 dt VB2O3(cid:2)SiO2
and l of the three regions. These are allowed to evolve with
3a
time from an initially small value within a numerical simula-
tion model. The evolution equations for l12 and derived from where subscript B2O3–SiO2 refers to the borosilicate present
flux and molar volumes are given below. The rate of change at the surface region (3-a), M refers to molecular weight, g
of l23 is given by the number of moles of MeO2 formed per the acceleration due to gravity, u the orientation of the sam-
unit area per unit time multiplied by the molar volume of ple with respect to gravity, and C refers to the rate
MeO2. The rate of formation of MeO2 is given by the total at which the borosilicate is added tBo2Ot3h(cid:2)eSiOsu2rface in moles per
flux of oxygen in through region 3–2, less than that used for unit areaperunit time.
oxidizing SiO(g) to SiO2 and CO(g) to CO2, which is 3/2 Finally, the net weight gained, weight of oxygen con-
times the SiO(g) flux in region 1–2 as given in Eq. 4. The sumed,and weightevaporated canbegiven as:
recession rate of MeB and SiC are simply related to the rate
2
of change of l through molar volumes and volume frac-
23
tions. Finally, the depleted zone length change is given by Wg¼l23ðfMeO2qMeO2þfgqgÞþl3aqg
othbetadinifftehreenfcoelloinwirnegceesqsiuoantiboentsw.een MeB2 and SiC. Thus, we (cid:2)RSiCfsqSiC(cid:2)RMeB2ð1(cid:2)fsÞqMeB2
5 ð1(cid:2)fÞ 3 f
W ¼ R s M þ R s M
O2 2 MeB2 V O2 2 SiCV O2
dl 3 2 1 X MeB2 Z t SiC R f
d2t3 ¼VMeO2ðjJO2(cid:2)32j(cid:2)2jJSiO(cid:2)12jÞ5f Wevap¼ Mspecies Jspeciesdtþ VSiC sMCO
MeO2 ðSiO2;SiO;B2O3Þ 0 SiC
dR dl V f
MeB2 ¼ 23 MeB2 MeO2 (11)
dt dt VMeO2ð1(cid:2)fsÞ (8)
dR V
SiC¼jJ j SiC The variation in time of the oxide scale thicknesses and
dt SiO(cid:2)12 fs the various weight gain/loss can be computed numerically
l ¼R (cid:2)R using Eqs. (1)–(11) using an evolutionary algorithm which is
12 SiC MeB2 started with a choice of, initially, arbitrarily small value for
l ,l ,and l .
12 23 3a
Inthe above set of equations,t is time, R is recession, V is
molar volume, and J is flux (moles per unit area per unit
time). (5) High-Temperature Regime
At high temperatures (likely above 2000 K) and/or at high
flow rates of ambient fluid (>1 m/s), where the evaporation
(4) External B O –SiO Glass Region (3-a) rates of SiO (g) and/or SiO(g) are sufficiently high, the exter-
2 3 2 2
The evolution with time for the thickness, l , of the external nal glassy layer will be lost and the glassy liquid region will
3a
glassy layer 3-a, is given by the rate of production of boria recede into the region 2–3, as shown in Fig. 1(b). Using 3i to
and silica less the rate of loss by evaporation at the surface identify the location of the receding glassy layer, the follow-
and the amount occupied by glass in region 2–3. From the ing equations describe the oxygen flux balance which fixes
literature, B O (g), SiO(g), and SiO (g) are known to be the the partial pressure of oxygen, P , at the interface i3i at
dominant ga2se3ous species in the B2O –SiO system in the the location3i, O2(cid:2)i3i
2 3 2
342 Journal of the American Ceramic Society—Parthasarathy et al. Vol. 95, No. 1
P (cid:2)P
J ¼P O2(cid:2)i3i O2(cid:2)i2f
O2(cid:2)3i2 O2(cid:2)B2O3(cid:2)SiO2 ql g
23
D P (cid:2)P
JO2(cid:2)a3i ¼ OR2T(cid:2)3ia105 O2ð(cid:2)1a(cid:2)qÞlO2(cid:2)i3ifg SinceJO2(cid:2)a3i ¼JO2(cid:2)3i2 (12)
23
qRTP P (cid:2)105qD P (cid:2)RTP P
P ¼ O2(cid:2)B2O3(cid:2)SiO2 O2(cid:2)i2 O2(cid:2)3ia O2(cid:2)a O2(cid:2)B2O3(cid:2)SiO2 O2(cid:2)i2
O2(cid:2)i3i qRTP (cid:2)105qD (cid:2)RTP
O2(cid:2)B2O3(cid:2)SiO2 O2(cid:2)3ia O2(cid:2)B2O3(cid:2)SiO2
The diffusive fluxes of gaseous species (SiO , B O , and sivity applied to the depleted region as well as the boundary
2 2 3
SiO) determine the rate of recession of the interface i3i. Tak- layer diffusion region at the surface for evaporation. The
ing the ratio of the depth of the glassy region to the length Knudsen effect on diffusivity was included as determined by
l of region 2–3 as q, the evolution equation for q is given the pore radius, assumed to be 0.5 lm in all predictions
23
as: shown herein; this choice was based on reported microstruc-
tures in the literature. The Knudsen effect only comes in
J ¼Dspecies(cid:2)3ia105Pspecies(cid:2)vap when the pores dry out under conditions where the evapora-
species(cid:2)vap RT ð1(cid:2)qÞl tion is sufficiently fast [Fig. 1(c)]. The Knudsen effect is also
(cid:6) 23
calculatedfor the depleted zoneas determined bySiCsize.
dq 1 1 3 2
dt ¼l23 fgfJSiO(cid:2)12VSiO2þðJO2(cid:2)3i2(cid:2)2JSiO(cid:2)(cid:7)12Þ5VB2O3 tial(Bp)ressBuroeriaofAocxtiyvgiteyn: atTthheecZalrcBul–aZtiorOn ofinetqerufialicberi(uim2)paanrd-
dl 2 2
(cid:2)J V (cid:2)J V g(cid:2)q 23 the partial pressures of volatile species at the outer surface
SiO2(cid:2)vap SiO2 B2O3(cid:2)vap B2O3 dt require the activity of boria in the borosilicate glass. The
(13) activity isotherm in the boria–silica system was measured at
1475 K and is found to be nearly ideal, although there is a
slightdeviationfromideality.28TheB O –SiO systemisalso
The right-hand side of the evolution equation is the rate 2 3 2
known for the lack of any phase separation.34 Based on
of production of B O and SiO less the evaporative flux; the
2 3 2 these, a unitactivity coefficientwas assumedin this work.
last term accounts for the increase in l during the time
23 (C) Glass Viscosity: The temperature-dependent vis-
increment. The weightgainof the sample is givenas:
cosity of boria was obtained from the works of Eppler and
Li et al.35,36 The data on viscosity of silica were reviewed by
Wg¼ql23ðfMeO2qMeO2þfgqgÞ (14) Doremus,37 and as per his conclusion, the data of Urbain
(cid:2)Rfq (cid:2)R ð1(cid:2)fÞq et al.38 were used for the temperature range 1400°C–2500°C
s s SiC MeB2 s MeB2 and that of Hetherington et al.39 for the temperature range
1400°C–1000°C. For compositional dependence of viscosity
Again, the variation in time of the oxide scale thickness
on boria content, a log-mean interpolation scheme was used
and the weight gain/loss can be computed using an evolu- consistent withsemiempirical models ofglass viscosities.40,41
tionary algorithm.
(D) Oxygen Permeability: The permeabilities of oxy-
gen in liquid boria and silica were obtained by fitting to data
from several sources. The boria data were obtained from
III. Model Predictions and Validation
Tokuda et al.,42 Luthra,43 and Schlichting.44 The oxygen per-
A numerical code written in Fortran was used for all the meability in silica was obtained from the works of Lamkin
model predictions shown in this work. The input variables et al.45 and Courtright.46 For the compositional dependence,
are the temperature–time history, environment parameters a log-mean approximation was used for interpolation consis-
(total pressure, fraction of oxygen), specimen orientation, tent with what might be expected from Stokes–Einstein rela-
length, fluid velocity, volume fraction of SiC, size of SiC par- tion as suggestedbyKarlsdottir and Halloran.17
ticles. All the thermodynamic and kinetic parameters taken The boria concentration at the surface layer must be lower
from the literature reside within the code; these include oxy- than that in the interior oxide + glass region due to evapora-
gen permeation and viscosity of silica and boria, equilibrium tive losses of boria being higher than silica. The surface layer
constants for all the reactions, and vapor pressures of the concentration depends on the evaporative rates, diffusion of
species. An infinitesimal layer of oxide and a glassy layer B in B-SiO , and the degree of convective mixing. This calcu-
2
were assumed present at an infinitesimal start time and the lation is beyond the scope of this work. Further, the authors
time evolution of all the parameters were calculated as dic- have found no experimental data on this. Hence, the surface
tated by the equations. An arbitrary time–temperature profile boria concentration was assumed to be a constant fraction of
could be thus simulated and the resulting thicknesses of the interior (interior B O concentration is ~0.72 for 20%
2 3
oxide, glassy layer, and depleted layer could be computed. SiC); this was the onlyloose parameter in the model andval-
From these, the recession, the weight gain, weight of oxygen ues of 0.8 and 0.9 (B O concentration of ~0.58 and ~0.65)
2 3
consumed, and weight of evaporated species could be com- for HfB and ZrB , respectively, gave the best correspon-
2 2
puted. dence.Theeffectiveporefractions,f ,forzirconiaandhafnia
p
were taken to be the same as was in ref. 24 for monolithic
diborideoxidation(0.03and0.04forHfO andZrO ,respec-
2 2
(1) Parameters tively). The aliovalent dopant concentration, C , in
dopant
A list ofvariables used inthe model isshownin Table I with MeO was taken to be less than 100 ppm, which permits the
2
a brief description. The model uses data from the literature neglect of oxygen permeation through the MeO phase (an
2
for all thermodynamic quantities, viz. equilibrium constants excellent assumption) (see ref. 22–24). The ambient fluid
and vapor pressures of species, which are available in the velocity was obtained from the literature when reported; it
compendium byBarin.32 varied from 0.0001 to 150 m/s, with the smaller values corre-
(A) Gas Diffusivity: The diffusivities ofgases ina mul- sponding to static air, and the higher values close to arc-jet
tigas solution were calculated using parameters given by conditions (behind the shock wave). The effect of fluid flow
Svehla,33 and the methods outlined in Ref. 22 The gas diffu- on evaporation rates was included in the model using a
January 2012 Oxidation Modeling of SiC-Containing Refractory Diborides 343
Table I. AList ofSymbolsUsed in This Work,with a BriefDescription andUnits
Symbol Units Description
f Effective volumefraction of poresin MeO that ispermeable to gas
p 2
f Volume fraction ofSiC
s
f Volume fraction ofMeO in the MeO –glassregion (2–3)
MeO2 2 2
f Volume fraction ofglass regionin the MeO –glass region(2–3)
g 2
a Activity ofspecies atinterface i2
species-i2
R m Recessionof MeB
MeB2 2
R m Recessionof SiC
SiC
V m3 Molarvolumes ofspecies
species
P atm Partialpressure of species
species
J mol/m2-s Fluxof speciesfrom interface 1–2
species-12
D m2/s Diffusivity ofspecies inregion 1–2
species-12
R J/mol-K Universal gas constant
P mol/m-s-atm Permeability coefficient ofoxygen in liquid borosilicate
I O2(cid:2)B2O3(cid:2)SiO2 m Depthof internal depletion(=R (cid:2) R )
12 s me
l m Thickness of zirconiaregionin the scale
23
I m Thickness of external glassylayer of B O –SiO (l)
3a 2 3 2
q m Thickness of zirconiascale overwhich B O –SiO (l) ispresent
2 3 2
T K Temperature
t s Time
M kg/mol Molecular weightof species i
i
ρ kg/m3 Density ofspecies i
i
g Pa-s Viscosity of speciesi
i
C mol/m2-s Rateof additionof boriato the external boriascale
JB2O3(cid:2)SiO2 mol/m2-s Rateof evaporation of species iat the external surface
species-vap
d m Boundarylayer thicknessfor surface evaporation
bdry
I m Lengthof specimen
specimen
V m/s Velocityof ambientfluid
fluid
W kg/m2 Netchange in weight perunit area
g
W kg/m2 Weightof O consumedperunit area
WO2 kg/m2 Weightof ev2aporated speciesperunit area
evap
boundary layer calculation (as detailed in ref. 22–24), but the
possible thinning of the external glassy layer from shear 1.E+04 SSiiCC --HsPin,t eorxeydg,e onx -yCgeons t:e Clloo,s Tterelloe,s Tsrleeer s[5s0le]r [50]
forces wasnotincluded in the model. n SC SiC-fast -oxygen-Costello, Treessler [50]
mi SC SiC -slow -oxygen-Costello, Treessler [50]
2m/ RCVamDb SeirCg ,O Wgbourrjei,l O[4p8i]la [54]
(D2a)ta Comparison of Model Predictions with Experimental K, npar 1.E+03 CSC iV ffaDacc eSe i -C-ZZ :hh Heenangrgr i eset t [ a5al3.l. ] [[5522]]
(A) SiC: There are a lot of oxidation data for SiC in nt,
the literature, and comparing the model to these data is a
st
important as a first check. As the oxygen permeability in sil- n
o
ica was used in the model, the comparison of model to data c
e
micsusotfbeoxtaidkaetnioans visalildimatiitoendobfythoexaysgseunmppteiromnetahbaitlittyheinkintehte- c rat 1.E+02
glassy phase, at least for SiC. Figure 2 shows a plot that oli
b
compares the available data for single and polycrystalline a
SiC in various forms reported by different investigations47–53 Par SiC
in pure oxygen; there is a large scatter in the data. Figure 2
1.E+01
includes data collected on high purity SiC by Ramberg
5.50E-04 6.00E-04 6.50E-04 7.00E-04
et al.,47 and Ogbuji and Opila53 as well as data obtained by
aCostelloandTressler49onSiCmadebydifferentprocessing 1 / Temperature, K
methods. The model is seen to fit fairly well with the poly-
Fig.2. Theoxidationkinetics,expressedasparabolicrateconstant,
crystal data, and the fast oxidizing (Si face) data for the sin-
in oxygen of single crystal and polycrystal SiC in various forms
gle crystal. reported by various investigators are shown compared with the
(B) SiC–ZrB2: A comprehensive work on the effect of model. The model prediction is shown as a solid line, whereas the
SiC content on the oxidation kinetics of UHTCs was per- experimentaldataareshownasdottedlines.
formed by Talmy.54 The study was conducted using a ther-
mogravimetric apparatus in flowing air. The weight gain of
samples exposed for 2 h in air is reported. In a more recent lines. The sample weight gainis the weight of the sample and
work, Wang et al.15 have conducted a similar study using oxidation products less the weight of external glassy phase
SiC volume fractions of 0, 5, 10, 15, and 20 vol% at temper- predicted to be lost by evaporation or viscous flow. The
atures up to1600°C for 1h in an air furnace. Figure 3 com- weight of oxygen consumed will be the total weight change if
pares the data with the model prediction for weight gain as a all of the oxidation products were retained on the sample.
function of SiC volume percent. Figure 3(a) shows data from Figure 3(c) includes time dependence data at 1200°C from
Talmy54 and Fig. 3(b) shows data from Wang et al.15 The Wang et al.The modelagreeswiththe trendsverywell.
model predictions for the sample weight gain and the total The temperature dependence of weight gain and scale
weight of oxygen consumed are shown as solid and dashed thickness data on a fixed composition of 20 vol%SiC–ZrB2
344 Journal of the American Ceramic Society—Parthasarathy et al. Vol. 95, No. 1
(a)
0.3 0.3 0.3
1473K, 2h 1573K, 2h 1773K, 2h
2Weight gain, kg/m00..12 DatafromTalmy[55] 2Weight gain, kg/m00..12 DatafromTalmy[55] 2eight gain, kg/m00..12 DatafromTalmy[55]
W
0 0 0
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
Volume % SiC Volume % SiC Volume % SiC
(b)
0.3 0.3 0.3
1473K, 1h 1673K, 1h 1873K, 1h
0.25 0.25 0.25
2m 2m 2m
kg/ 0.2 DatafromWangetal.[15] kg/ 0.2 DatafromWangetal.[15] kg/0.2
n, n, n,
ai0.15 ai0.15 ai0.15
G G G
ght 0.1 ght 0.1 ght 0.1
ei ei ei
W W W
0.05 0.05 0.05
DatafromWangetal.[15]
0 0 0
0 5 10 15 20 0 5 10 15 20 25 0 5 10 15 20 25
Volume%SiC Volume % SiC Volume % SiC
(c)
0.07
2m 0.06 1200C,ZrB2-15SiC
/
kg 0.05
n,
ai 0.04
G
t 0.03
h
g
ei 0.02
W DatafromWangetal.,[15]
0.01
0
0 100 200 300
Time,min
Fig.3. (a)TheeffectofvolumepercentSiContheoxidationweightgainin2hofUHTC(SiC–ZrB)samples,asmeasuredbyTalmy54shown
2
compared with the prediction of the model for three different temperatures. The solid lines are the predictions that include the flow of external
glassy layer under gravity and evaporation of boria or silica. The dotted lines show the predicted weight of oxygen consumed, which is the
maximum weight gain that the samples could have suffered from oxidation in the absence of flow/evaporation of the glassy layer. In (b), the
model is shown compared with data from a different investigation, by Wang etal.,15 which used a 1-h hold at temperature. In (c), the weight
gainasafunctionoftimeforaZrB–15vol%SiCsampleat1200°CreportedbyWangetal.15isshowncomparedwiththemodel.
2
in static laboratory air have been reported by Carney et al.10 contrast, Fahrenholtz 7 reported a depletion layer of 10 lm
This study took special care to document the effect of glass after 30 min at 1773 K in a ZrB –30% SiC sample; the
2
flow in two ways. They measured the weight changes of both model predicts only 0.6 lm for these conditions. For the
the sample and the crucible onto which some of the glass 20% SiC–ZrB case, the model predicts a depletion zone
2
had spread. They also measured the variation in scale thick- ranging from 1.8 lm at 1673 K to 3 lm at 1873 K which is
ness with sample surface orientation, and found significant closer to observations by Carney et al. Figure 5 shows a
scatter. Further, they reported on samples prepared by SPS comparison of the model with data for the time dependence
and hot press. For comparison with the model, the average of weight gain and scale thickness from two sources of
values of all the samples for a given condition were used. data.5,10 Data for hot-pressed samples and SPS-processed
The experimental data points attributed to Opila and Halbig samples and the sources of the data are distinguished with
in Fig. 4(a) were calculated from the parabolic constants for different symbol shapes. Once again, there is a reasonable
weight gain reported in their work.13 In Fig. 4(a), the total correspondence betweenthe dataand themodel.
weight gain (sample + crucible) is plotted along with the (C) SiC–HfB : Oxidation data on SPS-processed
2
model predictions, showing good correspondence. The plot high-density 20 vol%SiC–HfB samples have been collected
2
also shows that evaporation is significant above 1873 K. in static air from 1673 to 2273 K by Carney.11 In addition,
Figure 4(b) shows the average values for total scale (oxide the statistical variations from batch to batch and processing
+ glass) thicknesses and oxide thicknesses. Figure 4(b) route (hot-pressed versus SPS) have been studied for the
includes scale thicknesses data measured from images same composition at 1773 and 1873 K by Sevener.20
reported by Zhang et al.14 The data of Carney were obtained Figure 6(a) compares the data for weight gain with the
in an alumina furnace whereas the data of Zhang et al. were model predictions. The data for weight gain include separate
obtained using a zirconia furnace. A clear internal depletion measurements for the sample (open symbols) and crucible
layer was not observed/reported in the experiments. In + sample (filled symbols). The model predictions include
January 2012 Oxidation Modeling of SiC-Containing Refractory Diborides 345
(a) (a)
1.E+00 0.2
1873K-Carney et al.[10] -SPS
2.5h, S(cid:415)llAir Carneyetal.[10] SPS 1873K-Carney et al.[10] -HP
Carneyetal.[10] HP
2 2 1873K -Model
g/m WtO2 OMpoildae,lHaWlbtige[v1a3p] g/m 0.15 11690000KK --MLeovdineel et al.[5]
k Model WtGain k
n, Model WtO2 n, 1600K -Levine et al.[5]
gai 1.E01 ai 0.1
g
ht t
g h
ei g
W ei
0.05
Wtgain W
Wtevap
1.E02
4.5E04 5.5E04 6.5E04 0
0 50 100 150 200 250 300
1/Temperature,K
Time (min)
(b)
2000 1818 1666 1538K
(b)
1.E03
TotalScale Oxide:Carney[10] (aveHP,SPS) 1.4E-04 1873K -Oxide-Carney et al.[10] -HP
1900K -oxide -Levine et al.[5]
(oxide+glass) Total:Carney[10](AveHP,SPS) 1873K -Oxide+glass -Carney et al.[10]-HP
m Oxide:Zhangetal.[14] m 1.2E-04 1873K -oxide+glass-Carney et al.[10]-SPS
ness,1.E04 Total:Zhangetal.[14] ess, 1.0E-04 111998007003KKK ---dOmexopiddleeetl+i-ogonlx a-isdLse-eLveinvein eet eatl .a[5l.][5]
k n 1873K -model-oxide+glass
hic OxideScale ck 8.0E-05 1873K -model-depletion
T hi
ale1.E05 e t 6.0E-05
Sc al
c 4.0E-05
S
deple(cid:415)on
2.5h,S(cid:415)llAir
2.0E-05
1.E06
4.5E04 5.5E04 6.5E04 0.0E+00
1/Temperature,K 0 100 200 300
Fig.4. (a) Oxidation weight gain in ZrB–20vol%SiC samples, Time, min
2
measured in static air after 2.5h of exposure, by Carney etal.10
showncomparedtothemodelpredictions.ThedatafromOpilaand Fig.5. (a) Model compared with oxidation weight gain measured
Halbig13 are also shown. The solid lines are model predictions for as a functionof time duringoxidation of ZrB–20vol%SiC samples
2
weightgain,weightofoxygenconsumed,andweightevaporated.(b) in static air at 1873K, by Carney etal.10 and at 1900 and 1600K
Average of reported values for the oxide scale thickness (open by Levine etal.5 The solid lines are model predictions for the two
symbols) and total (oxide+glass) scale thickness (filled symbol) for temperatures. (b) Model compared with reported oxide thickness
the same exposures; values measured on different sides of the (open), total (oxide+glass) thickness (filled), and internal depletion
sampleswereaveragedforthisplot.ValuesfromtheworkofZhang thickness (dash) as marked. In both (a) and (b), the squares
etal.14 which used a zirconia furnace, are also included. The solid represent Carney’s hot-pressed samples, the diamonds Carney’s SPS
lines are predictions for total scale, oxide scale, and depletion samples,andtrianglesdatafromLevineetal.
thicknesses. No depletion thickness was reported, but partial
depletionwasnoticedunderFIB/TEMinvestigation.10
was detected. Enhanced oxidation of SiC due to the presence
total weight of oxygen consumed, sample weight gain, and ofimpuritiesiswellknown(forexample,seeRamberget al.47)
weight evaporated. Evaporation is predicted to dominate at Figure 7 compares the model predictions for weight gain
around 2000 K, and the model is consistent with the drop in as a function of time for 20 vol%SiC–HfB in furnace air at
2
sample weight gain at this temperature. In general, the model 1773 and 1873 K, with data reported by Sevener.20 The data
captures the trends well except at the highest temperature. In include SPS-processed material and hot-pressed material with
Fig. 6(b), the scale thicknesses measured are compared with different initial SiC particle sizes obtained from different pro-
the model. The microstructures of the oxidation product cessing routes. The data reported for the combined weight
were complex at the highest temperatures, and there was gain of sample and crucible were used. The correspondence
ambiguity in the definition of the depleted zone. Thus, the is once again found to be good, although there is consider-
sum of oxide scale and depletion zone (open symbols) was able scatter in the data. The model prediction plotted in
used for this comparison. The total scale thicknesses (filled Fig. 7 includes the total weight of oxygen consumed showing
symbols) are also plotted. The correspondence between the thatevaporative loss becomes significantat1873 K.
model and data is seen to be reasonable, except for the data (D) Arc-Jet Tests on SiC–MeB : The expense of con-
2
at the highest temperature reported, viz. 2173 K, which devi- ducting arc-jet tests has limited the number of investigations
ates significantly from the model. However, these data also and the extent of data available on oxidation kinetics during
deviate significantly from the extrapolation of the data at these tests. However, data are available for a few SiC–MeB
2
lower temperatures. On further examination of the sample, compositions. Montverde and Savino have tested a hemi-
significant contamination of alkali elements (mainly Ca) pos- spherical sample of ZrB –15 vol%SiC under arc-jet condi-
2
sibly from the furnace or crucible (Ca-stabilized zirconia) tions where the surface temperature reached ~2193 K for
346 Journal of the American Ceramic Society—Parthasarathy et al. Vol. 95, No. 1
(a) 2222 2000 1818 1666 K (a)
0.15
1.E+00
1773K
2m (HotPressed)
Sample+Crucible
/
g 0.1
2 K WtO
m n, 2
g/ 1.E01 gai
n,k ght 0.05
gai Wt.O2 Wei Wtgain
t
gh 1.E02 Sample 0
Wei Wt.Gain 0 100 200 300 400
Time,min
Wt.Evaporated
(b)
0.15
1.E03
1873K
4.0E04 5.0E04 6.0E04 7.0E04
2m (SPSprocessed)
1/Temperature,K Kg/ 0.1 WtO2
n,
ai
(b) 2222 2000 1818 1666 K g Wtgain
1.00E03 ght 0.05
ei
W
m 0
s, 1.00E04 0 100 200 300 400
s
e TotalScale Time,min
n
k (c)
hic Oxide 0.15
T +deple(cid:415)on 1873K
ale 1.00E05 2m (HotPressed)
c /
S Kg 0.1 WtO2
Deple(cid:415)on n,
ai
g
1.00E06 ght 0.05
ei
4.00E04 5.00E04 6.00E04 7.00E04 W
Wtgain
1/Temperature,K
0
Fig.6. Comparision of model with experimental data on oxidation
kinetics of HfB–20vol%SiC in the high-temperature regime (up to 0 100 200 300 400
2 Time,min
2173K)instaticairshowingweightgainin(a)andscalethicknesses
in (b). All the data are from the work of Carney.11 The solid lines Fig.7. Oxidationdataondifferentbatchesofhot-pressed(aandc)
are model predictions. In (a), both sample weight (open) and and SPS-processed samples (b) of 20vol%SiC–HfB composition,
sample+crucible (filled) weights are shown, indicating flow is collectedat1773and1873KbySevener.20Thesolidl2inesaremodel
significant above 2000K. Similarly in (b), total scale thickness predictions for the weight gain and weight of oxygen consumed; the
(filled) and oxide scale thickness (open) are shown. Due to differencearisesfromevaporationofvolatileoxides.
experimental ambiguity in demarcation between depletion and oxide
scale, oxide scale+depletion thickness are shown for data and
model.
temperature of 1803 K, and another at 1963 K. The model
predictions forboth these temperatures are shown.
In general, the model underpredicts the kinetics for experi-
325 min.55 From their metallographic images along with mental data on samples exposed to arc jet. The model is
EDS maps, the thicknesses of the oxide layer, glass layer, much closer at lower temperatures. The data from furnace-
and depletion layer were obtained. These are plotted along exposed samples show slower kinetics and are in reasonable
withmodelpredictionsinFig. 8(a).Abilinearapproximation correspondence withthe model.
of the measured temperature–time profile, used as input for
the model, is shown as inset. More recently, Savino et al.
IV. Discussion
have conducted a similar study at higher surface tempera-
tures using the arc jet.56 From the reported thermal history (1) Model Strength and Weaknesses
and calculated scale thicknesses from the published micro- A model has been presented to interpret the oxidation kinet-
graphs, the plot shown in Fig. 8(b) was generated which ics of SiC-containing refractory metal diborides. The model
shows the modelpredictions comparedto thedata. includes the effect of viscous flow of the outer glassy scale
Data on arc jet tested samples of SiC–HfB have been and evaporative losses of volatile species under a boundary
2
reported on by Carney11 and Gasch et al.12 In particular, layer condition established by ambient fluid flow velocity.
Carney reported on scale thicknesses for both furnace- The model predicts the total weight of oxygen consumed,
exposed and arc-jet-exposed samples of the same batch of sample weight gain, weight of evaporated species, oxide
sample under the same temperature (1773 K) and duration. thickness, glass thickness, depletion layer thickness, and
Figure 8(c) shows the model predictions compared with fur- recession of the substrate. The model uses as input parame-
nace data and the arc-jet data. Figure 8(d) shows the data ters, the exposure temperature and time or a thermal profile,
from Gasch et al., who conducted two tests: one at surface ambient oxygen partial pressure, gas chemistry, and fluid