March 1998 Kinetics of Thermal, Passive Oxidation of Nicalon Fibers wheree= kd and F= wh are constants under a constant x temperature. Solving Eq. (17)and rearranging the result yield (ER+ 2F)u+2ER(n R-In u)? 2R-x= 2R (30) Rt+M (19) and Substituting the initial condition(Eq(18))into Eq. 19)yields R Substituting Eq(20)into Eq. (19)and rearranging teA. (0) Substituting Eqs. (29)and(30)into Eq(22)and rearranging M=(ER+2F+4)R2 he result yields Gt layer, which means that oxidation is controlled by the rate of where= 4kC N is a constant under a constant temperature chemical reaction Substituting Eq. (1)into Eq. (21)and rearranging the result For the oxidation of a flat plate, the diffusion of oxygen will yields the general relationship for the oxidation kinetics of control the reaction kinetics at a large oxide thickness, which cylindrical fibers yields a parabolic relationship between the oxide thickness x and time t. However, for the oxidation of a cylindrical fiber, the 2F(2-)x+E(2R-x)x+2(R-x2h area of interface at which the oxidation occurs decreases as x ncreases. The shrinking reaction interface will affect the gen (22) eral oxidation kinetics, especially when x approaches the fiber here E, F, and G in Eq (22)are to be determined from fitting radius. When x approaches R (i.e, x-R), the following ap- he experimental data. They are functions of temperature and proximation can be made remain constant for a fixed temperature Many researchers prefer to use an a parameter, which is the (33) fractional reacted volume. instead of oxide thickness. x. to describe oxidation kinetics. 9-1 To compare our results with 2R-xgR their work, we can also write Eq(22)in terms of a, which can be calculated as and (R-x)2ln(1 0 Solving Eq(23)for x yields Substituting Eqs. (33)(35)into Eq.(22) and rearranging the result vield Substituting Eq (24)into Eq (22) yields 2F+ER+4 2FRo+ERa+(1-a)In(1-a) Equation(36)describes a linear rela hip between the oxide +4R[-(1-a)2]=Gt (25) thickness x and time t when x approaches the fiber diameter R, which indicates that the general Equation(22)can be used to describe the oxidation kinetics trolled by the shrinking area of the n kinetics are con- infinity(R-oo), the fiber surface will become essentially flat oxygen diffusion ng area of the and Eq.(22)should approach the oxidation kinetics of a flat plate, as derived by Deal and Grove. 10 This is indeed the case, Ill. Comparison with Experimental Results s demonstrated below. As R approaches infinity, the following and discussions approximation can be made Oxidation experiments were performed on individual (26) NicalonTM fibers(Dow Corning, Midland, MI) that had been separated from a tow. The fibers were heated in a partial ox gen pressure of 0. 14 atm(-0.014 MPa)at a heating rate of 200%C/h to 1200%C. held for 4.8. or 16 h and then furnace cooled to room temperature. A high-resolution scanning elec tron microscopy(SEM)microscope(Model 6300FVX, JEOL, Tokyo, Japan) was used to measure the fiber diameters and the Equation(26)originates from the fact that (R-r)In(I-x/ through a service contract to maintain its accuracy in magni- →-xasR→∞.S and rearranging th ing Eqs. (26)and(27)into Eq ( 22) fication. The measurement was performed directly using the SEM microscope instead of the SEM micrograph print. Before Ex2+4(F+1)x-Gt=0 measurement of the fiber diameter and oxide thickness the sample stage was tilted so that the fiber cross section was Equation(28)is identical to the equation derived by d almost perpendicular to the electron beam, which minimized Grove o for a flat plate when the initial oxide thickness Another special case for Eq.(22)is at the beginning c he possible error caused by tilting the cross section. The fiber diameter and oxide thickness were taken as being the average xidation, when the oxide layer thickness x is approximately of four measurements on each cross section. One cross section zero. The following approximations can be made in this situ- was examined per fiber. Figure 2 shows a typical SEM micro- graph of the cross section of a fiber with an oxide layerwhere E = k/D and F = k/h are constants under a constant temperature. Solving Eq. (17) and rearranging the result yields ~ER + 2F!u2 + 2ER~ln R − ln u!u2 + 4uR = S− 4kC* N DRt + M (19) Substituting the initial condition (Eq. (18)) into Eq. (19) yields M = ~ER + 2F + 4!R2 (20) Substituting Eq. (20) into Eq. (19) and rearranging the result yields 2F~R2 − u2 ! + ERSR2 − u2 + 2u2 ln u RD + 4R~R − u! = GRt (21) where G 4 4kC*/N is a constant under a constant temperature. Substituting Eq. (1) into Eq. (21) and rearranging the result yields the general relationship for the oxidation kinetics of cylindrical fibers: 2FS2 − x RDx + EF~2R − x!x + 2~R − x! 2 ln S1 − x RDG + 4x = Gt (22) where E, F, and G in Eq. (22) are to be determined from fitting the experimental data. They are functions of temperature and remain constant for a fixed temperature. Many researchers prefer to use an a parameter, which is the fractional reacted volume, instead of oxide thickness, x, to describe oxidation kinetics.19–21 To compare our results with their work, we can also write Eq. (22) in terms of a, which can be calculated as a = pR2 l − p~R − x! 2 l pR2 l (23) Solving Eq. (23) for x yields x=R[1 − (1 − a) 1/2] (24) Substituting Eq. (24) into Eq. (22) yields 2FRa + ER2 [a + (1 − a) ln (1 − a)] + 4R[1 − (1 − a)1/2] 4 Gt (25) Equation (22) can be used to describe the oxidation kinetics of a cylindrical fiber with a diameter R. When R approaches infinity (R → `), the fiber surface will become essentially flat and Eq. (22) should approach the oxidation kinetics of a flat plate, as derived by Deal and Grove.10 This is indeed the case, as demonstrated below. As R approaches infinity, the following approximation can be made: ~R − x! ln S1 − x RD ≈ −x (26) and x R ≈ 0 (27) Equation (26) originates from the fact that (R − x) ln (1 − x/R) → −x as R → `. Substituting Eqs. (26) and (27) into Eq. (22) and rearranging the result yields Ex2 + 4(F +1)x − Gt 4 0 (28) Equation (28) is identical to the equation derived by Deal and Grove10 for a flat plate when the initial oxide thickness is zero. Another special case for Eq. (22) is at the beginning of the oxidation, when the oxide layer thickness x is approximately zero. The following approximations can be made in this situ￾ation: ln S1 − x RD ≈ − x R (29) 2R − x ≈ 2R (30) and R − x ≈ R (31) Substituting Eqs. (29) and (30) into Eq. (22) and rearranging the result yields x = Gt 4~F + 1! (32) Equation (32) indicates a linear relationship between the oxide thickness x and time t under the condition of a very thin oxide layer, which means that oxidation is controlled by the rate of chemical reaction. For the oxidation of a flat plate, the diffusion of oxygen will control the reaction kinetics at a large oxide thickness, which yields a parabolic relationship between the oxide thickness x and time t. However, for the oxidation of a cylindrical fiber, the area of interface at which the oxidation occurs decreases as x increases. The shrinking reaction interface will affect the gen￾eral oxidation kinetics, especially when x approaches the fiber radius. When x approaches R (i.e., x → R), the following ap￾proximation can be made: 2 − x R ≈ 1 (33) 2R − x ≈ R (34) and ~R − x! 2 ln S1 − x RD ≈ 0 (35) Substituting Eqs. (33)–(35) into Eq. (22) and rearranging the result yields x = Gt 2F + ER + 4 (36) Equation (36) describes a linear relationship between the oxide thickness x and time t when x approaches the fiber diameter R, which indicates that the general oxidation kinetics are con￾trolled by the shrinking area of the reaction interface, instead of oxygen diffusion. III. Comparison with Experimental Results and Discussions Oxidation experiments were performed on individual Nicalon™ fibers (Dow Corning, Midland, MI) that had been separated from a tow. The fibers were heated in a partial oxy￾gen pressure of 0.14 atm (∼0.014 MPa) at a heating rate of 200°C/h to 1200°C, held for 4, 8, or 16 h, and then furnace cooled to room temperature. A high-resolution scanning elec￾tron microscopy (SEM) microscope (Model 6300FVX, JEOL, Tokyo, Japan) was used to measure the fiber diameters and the oxide thickness. The SEM microscope was calibrated regularly through a service contract to maintain its accuracy in magni￾fication. The measurement was performed directly using the SEM microscope instead of the SEM micrograph print. Before measurement of the fiber diameter and oxide thickness, the sample stage was tilted so that the fiber cross section was almost perpendicular to the electron beam, which minimized the possible error caused by tilting the cross section. The fiber diameter and oxide thickness were taken as being the average of four measurements on each cross section. One cross section was examined per fiber. Figure 2 shows a typical SEM micro￾graph of the cross section of a fiber with an oxide layer. March 1998 Kinetics of Thermal, Passive Oxidation of Nicalon Fibers 657