Journal of the American Ceramic SocietyJacobson et al. Vol. 82. No 6 The factor of three in the last term is due to the three boron 120011001000900 atoms in H3 B3 O(g). The diffusivities in Eqs. (5H9)above are effective diffusivities, composed of a molecular and a Knudsen 10-12 component. The molecular diffusivity of the H, B, O(g)in the BN/B2037 annular channels can be estimated using the standard Chap- Shaded area--regio man-Enskog correlation: 25 of B203 stability 0.001853742 D molec 10-20 Sic/SiO2-7 (UoH-B. O).Bo(Mo,MH,B (10) Here T is the absolute temperature, P the total pressure, and M BN/B203 he molecular weight of the subscripted species. The parameter is the average collision diameter, which is equal to the average of the molecular diameters of o, and the H-B o (g) molecules. The molecular diameters for the h, B, o(g) molecules are taken from similar molecules.26 The collision 1032 integral, OLo, -HBO, is determi tables of average inter- molecular force meters 二时 ending collision inte- als. 25,26 Tables II(A)an the calculated mole 0.0006 0.0007 0.0008 0.0009 diffusivities for each of the H, B O(g)species. At hannel diameters of0.5 and 1.0 um, it is important to consider the knudsen diffusivity contributions. The Knudsen diffusion coefficient, D,, is given by Fig 8. Calculated oxygen potentials at BN/B,O3 and SiC/SiO,(Un- derline means the activity of that element is unity. (11) on time, which is not observed. Similarly, chemical-reaction control would lead to linear rates. It is likely that diffusion in It should be noted (Il) is for a cylind not al the pores with concurrent pore closure is rate controlling annulus. However In this This process of BN volatilization is analogous to carbon, T, and M as given above. The two diffusivities are combined meter with the same fibers:e 21-23 BN volatilization via B2O, to H, B, o (g)is some- ere are several treatments of this process in the lit what simpler, because the formation of the volatile species is a (12) one-step process, as opposed to carbon volatilization, which is Volatilization of BN occurs via the three vapor species dis- Tables II(A)and(B)list the Knudsen and effective diffusivities cussed above. First consider the flux due to each species: The complication is that these volatile species do not come HxByO-HrB, O- HrBy directly from the bn, but rather as shown in Fig. 11. So the reactions are This is simply Fick's first law, with the effective diffusivity BN +=O2(g)=B2O3()+n2(g) (13a) the pressure gradient through the pore VBN the molar vol- B2O3+H,O=2HBO(g) (13b) f Bn. and p apor pressure of H, B, o(g) Now we can convert Eq. (5)to a recession distance for B2O3+3H,0=2H, BO3(g) (13c) each H, B, O- specie dyH,B E=-J/H-B, O_ BN=BN/9 dp 3B2O3+3H2O=2H3B3O6g) (13d dt rt dyH-,o (6) Note that B,O is at less than unit activity. Recent data on the B,O3-SiO2 system indicate that the activities can be approxi We can approximate the pressure gradient as mated as ideal. However, we do not know the composition of the borosilicate melt, and, furthermore, it changes as the dPH,B, o, APH, B, 0: PH,, O reaction progresses. However, as an approximation, we shall assume the mole fraction of B, O3 to be 0.5 and take the activ ty of B,, to be 0.5. Table Ill lists the vapor pressures of Here PH, B o, is the pressure of the particular H, B,O-g)spe- HBO2 (g), H3 BO3(g), and H3 B3 O(g)in equilibrium with cies and yH. bo the recession distance due to this species. Now B2O(e)with an activity of 0.5. These pressures are used in the 2V PHBon Equation( 8)indicates that recession rates are parabolic with RI (8) time. However, our data suggest the annular channels seal as time progresses and limit the outward flux. This is probably The total recession rate of BN, which was measured, is simply due to oxidation of the annular channel walls the sum of the recession rates due to each H, B, o(g)species The growth of SiO, on SiC is best described by a linear- parabolic equation from Deal and Grove yg Rr(HBO2HBO2*DHaBO3H3BO3 x2+Ax= B(t+T) (14) 3B306 H3 B306 Here B is the parabolic rate constant, B/A the linear rate con-on time, which is not observed. Similarly, chemical-reaction control would lead to linear rates. It is likely that diffusion in the pores with concurrent pore closure is rate controlling. This process of BN volatilization is analogous to carbon￾interphase burnout, observed in composites with carbon-coated fibers. There are several treatments of this process in the lit￾erature.21–23 BN volatilization via B2O3 to HxByOz(g) is some￾what simpler, because the formation of the volatile species is a one-step process, as opposed to carbon volatilization, which is a two-step process.21–23 Volatilization of BN occurs via the three vapor species dis￾cussed above. First consider the flux due to each species: JHxByOz = −DHxByOz dcHxByOz dyHxByOz = −DHxByOz RT dPHxByOz dyHxByOz (5) This is simply Fick’s first law, with the effective diffusivity DHxByOz calculated below. J is the flux of HxByOz(g), R the gas constant, y the recession distance, dPH xBy Oz /dyH xBy Oz the pressure gradient through the pore, VBN the molar vol￾ume of BN, and PHxByOz the vapor pressure of HxByOz(g). Now we can convert Eq. (5) to a recession distance for each HxByOz species:24 dyHxByOz dt = −JHxByOz VBN = VBNS DHxByOz RT dPHxByOz dyHxByOz D (6) We can approximate the pressure gradient as dPHxByOz dyHxByOz = DPHxByOz DyHxByOz ≈ PHxByOz yHxByOz (7) Here PHxByOz is the pressure of the particular HxByOz(g) spe￾cies and yHxByOz the recession distance due to this species. Now we can integrate Eq. (7): yH xByOz 2 = 2VBN~DHxByOz PHxByOz t! RT = kHxByOz t (8) The total recession rate of BN, which was measured, is simply the sum of the recession rates due to each HxByOz(g) species: yB = 2 VBNt RT ~DHBO2 PHBO2 + DH3BO3 PH3BO3 + 3DH3B3O6 PH3B3O6 ! (9) The factor of three in the last term is due to the three boron atoms in H3B3O6(g). The diffusivities in Eqs. (5)–(9) above are effective diffusivities, composed of a molecular and a Knudsen component. The molecular diffusivity of the HxByOy(g) in the annular channels can be estimated using the standard Chap￾man–Enskog correlation:25 Dmolec = 0.001853T1/2 P~sO2–HxByOz ! 2 VO2–HxByOz S 1 MO2 + 1 MHxByOz D (10) Here T is the absolute temperature, P the total pressure, and M the molecular weight of the subscripted species. The parameter sO2–HxByOz is the average collision diameter, which is equal to the average of the molecular diameters of O2 and the HxBy￾Oz(g) molecules. The molecular diameters for the HxByOz(g) molecules are taken from similar molecules.26 The collision integral, VO2–HBO2 , is determined from tables of average inter￾molecular force parameters and corresponding collision inte￾grals.25,26 Tables II(A) and (B) list the calculated molecular diffusivities for each of the important HxByOz(g) species. At channel diameters of 0.5 and 1.0 mm, it is important to consider the Knudsen diffusivity contributions. The Knudsen diffusion coefficient, Dk, is given by25 Dk = 2 3 d S 8RT pM D 1/2 (11) It should be noted that Eq. (11) is for a cylindrical pore, not an annulus. However, for this approximation, it is adequate. In this equation, d is the pore diameter with the same definitions for R, T, and M as given above. The two diffusivities are combined as22 1 DHxByOz = 1 Dk + 1 Dmolec (12) Tables II(A) and (B) list the Knudsen and effective diffusivities calculated for the experimental conditions. The complication is that these volatile species do not come directly from the BN, but rather via a borosilicate intermediate, as shown in Fig. 11. So the reactions are BN + 3 2 O2~g! = B2O3~l! + N2~g! (13a) B2O3 + H2O = 2HBO2~g! (13b) B2O3 + 3H2O = 2H3BO3~g! (13c) 3B2O3 + 3H2O = 2H3B3O6~g! (13d) Note that B2O3 is at less than unit activity. Recent data on the B2O3–SiO2 system14 indicate that the activities can be approxi￾mated as ideal. However, we do not know the composition of the borosilicate melt, and, furthermore, it changes as the reaction progresses. However, as an approximation, we shall assume the mole fraction of B2O3 to be 0.5 and take the activ￾ity of B2O3 to be 0.5. Table III lists the vapor pressures of HBO2(g), H3BO3(g), and H3B3O6(g) in equilibrium with B2O3(,) with an activity of 0.5. These pressures are used in the flux calculations. Equation (8) indicates that recession rates are parabolic with time. However, our data suggest the annular channels seal as time progresses and limit the outward flux. This is probably due to oxidation of the annular channel walls. The growth of SiO2 on SiC is best described by a linear– parabolic equation from Deal and Grove:27 x2 + Ax 4 B(t + t) (14) Here B is the parabolic rate constant, B/A the linear rate con￾Fig. 8. Calculated oxygen potentials at BN/B2O3 and SiC/SiO2. (Un￾derline means the activity of that element is unity.) 1478 Journal of the American Ceramic Society—Jacobson et al. Vol. 82, No. 6