Effects of Non-Planar Geometries and Volumetric Expansion in the Modeling of Oxidation in Titanium

Transcription

1 Effects of Non-Planar Geometries and Volumetric Expansion in the Modeling of Oxidation in Titanium Pavlin B. Entchev Dimitris C. Lagoudas John C. Slattery January 12, 2000 Center for Mechanics of Composites Department of Aerospace Engineering Texas A&M University College Station, TX Abstract Development of a model for oxidation of titanium based upon diffusion of oxygen in the oxide is presented in this work. Equations describing motion of the oxygenoxide and metal-oxide interfaces are derived using jump mass balance. The volumetric expansion of the oxide and its impact on the motion of interfaces is fully accounted for. An analytical solution is obtained for the case of one-dimensional planar oxidation and is used to estimate the diffusion coefficient of oxygen into titanium dioxide. Numerical results are presented for the cases of oxidation on the surface of a cylinder, oxidation of a cylindrical hole and oxidation of a sphere and comparisons are made with experimental data. 1 Introduction Titanium (Ti ) made components are often exposed to harsh environments, such as high humidity, elevated temperature and presence of chemically active substances. A considerable effort has been made to minimize the effect of the environment on the structural components using protective coatings. However, the total elimination of these effects does not appear to be a valuable solution. A number of researchers have studied the oxidation mechanisms of Ti under a variety of conditions [1, 2, 3, 4, 5]. A class of oxidation models has been proposed in the works [6, 5]. An extensive study on the oxidation of titanium is given in [7] with its theoretical part based on the well known diffusion-based model by Wagner [6]. Mathematical description of diffusion problems involving moving boundaries is given in the works of Crank [8, 9]. Author to whom correspondence should be addressed; Phone: ; Fax: ; lagoudas@tamu.edu 1

2 A theoretical description of oxidation of titanium is presented in [10], with its emphasis on the mathematical derivation of a two-dimensional (2D) model and its analytical solution for some special cases. This model accounts for the immobilized oxygen, contained in the oxide, by introducing a jump in the oxygen concentration and for the oxygen dissolved in Ti, but it does not account for the volumetric expansion of TiO 2. Numerical solutions for different 2D geometries have been presented in [11, 12, 13]). Although there have been many models used to simulate oxidation of different material systems, many of them share common characteristics. For example, researchers have recently investigated modeling of oxidation of silicon [14, 15]. Typically, these models account for the diffusion of O 2 through the oxide layer and the associated reaction at the metal-oxide interface. The models developed by the above mentioned researchers have been widely used to simulate the oxide layer growth on various geometries. However, they all possess one major inconsistency with the observed experimental data, namely, they do not consider expansion (or shrinking, in some cases) of the oxide [16, 17]. This effect is measured by the Pilling- Bedworth ratio (PBR), defined as ratio of the molar density of the metal to the molar density of the oxide (values of PBR for some metals are given in the classical work of Pilling and Bedworth [16]). If the value of PBR is greater than one (which is the case of Ti ) there is volumetric expansion as a result of the formation of the oxide while if the value of PBR is less than one the oxide shrinks. The reader should be aware that the expansion of the oxide is observed not only during the oxidation of Ti, but is a common characteristic for most of metals. There is a number of papers devoted to the effect of the expansion of the oxide (see, for example, [18, 17, 19, 20]). However, most of the existing work is experimental. Therefore, the main objective of this work is to develop a model of oxidation of Ti, which takes into account the expansion of TiO 2. For simplicity, it is assumed that Ti is oxygen free, that is, any effects of oxygen dissolution in Ti are neglected. In addition, the effects of stress on the diffusion are omitted. For detailed description of the coupling between the mechanical stress and the diffusion process the interested reader is referred to studies [21] and [22]. The remainder of the paper is organized as follows: the next section is devoted to the derivation of three-dimensional (3D) model of oxidation of Ti ; section 3 describes special one-dimensional cases of the derived model; results and discussion are presented in section 4 and conclusions are made in section 5; finally, the comparison between the proposed model and the currently existing oxidation models is presented in the Appendix. 2 Model Development for Oxidation of Titanium Three-dimensional oxidation of a Ti specimen is considered. The schematic of the problem is shown in Fig.1. As it can be seen on Fig.1, there are three different regions defined: Region I, corresponding to the surrounding oxygen environment; Region II, corresponding to TiO 2 ; and, Region III, corresponding to Ti. Also defined are two interfaces: I-II interface between Regions I and II and II-III interface between Regions II and III. During oxidation, oxygen penetrates through the I-II interface into Region II and diffuses into TiO 2. As soon as O 2 reaches II-III interface the oxidation reaction takes place and TiO 2 forms, causing both 2

3 w (1) O 2 w (2) v ( II ) ( TiO2 ) TiO 2 +O 2 v ( III ) Ti+O 2 Figure 1: Schematic of oxidation of Ti. I-II and II-III interfaces to move; the latter due to the consumption of Ti during the oxidation and the former due to the expansion of TiO 2. In addition, the expansion induces motion of material particles of both TiO 2 in Region II and Ti in Region III. Based on the above problem definition the following field variables, which are functions of time and position, are sufficient in the modeling of oxidation: concentration (molar density) of O 2 in Regions I, II and III, concentration of TiO 2 in Region II, concentration of Ti in Region III, velocity of TiO 2 particles in Region II, velocity of Ti particles in Region III and velocities of both I-II and II-III interfaces. The following assumptions are made in the derivation of the model: 1. The reaction at the II-III interface is instantaneous and is given by Ti + O 2 TiO 2 ; (1) 2. Ti moves as a rigid body, i.e., v (III) = const; 3. Concentration of TiO 2 is constant; 4. Thermodynamic equilibrium is established at I-II interface; 5. Ti is initially oxygen-free; 6. The system is isothermal, which means that the energy released by the reaction is dissipated rapidly; 7. All physical parameters are assumed to be constants. 3

4 In this work, the concentration of a given species is defined as a ratio of its density to its molecular weight. Assumptions 1 and 5 imply that the concentration of O 2 at the II- III interface is equal to zero as well that it remains zero in Ti for all subsequent time. In addition, since equilibrium has been assumed at I-II interface, the concentration of O 2 at the interface is known. This also implies that it is not necessary to solve for the concentration of O 2 in the Region I. Therefore, from the field variables introduced above, the following ones remain to be determined: c (II) - concentration of oxygen in Region II; v (II) - velocity of TiO 2 in Region II; w (1) - velocity of the I-II interface (interface 1); w (2) - velocity of the II-III interface (interface 2). The approach taken in this work is to write differential mass balance for all species in Region II (since mass balance for Regions I and III is trivial) and the jump mass balance for I-II and II-III interfaces. Using the Eulerian description, the differential mass balance for Region II is: O 2 : TiO 2 : c (II) t c (II) t + divn (II) = 0; (2) + divn (II) = 0, (3) where N (II) and N (II) are the molar fluxes of O 2 and TiO 2 in Region II in the laboratory frame of reference. Using the definition of the molar flux of TiO 2 N (II) c (II) v(ii), (4) and assumption 3 one can conclude that the mass balance for TiO 2 is expressed by divv (II) = 0. (5) An expression for the molr flux N (II) is derived using Fick s first law for binary diffusion: N (II) = c(ii) c (II) ( N (II) + N (II) ) c (II) D (O2,TiO 2 ) c(ii) c (II), (6) where c (II) is the total concentration, i.e., c (II) = c (II) + c (II). Further, after some simplifications and using eq.(4), the above expression for the molar flux of O 2 in Region II becomes: N (II) = c (II) v(ii) D (O2,TiO 2 ) c (II). (7) 4

5 Substituting eq.(7) into the expression for the mass balance of O 2 in Region II (eq.(2)) yields: c (II) t D (O2,TiO 2 ) c (II) + v (II) c (II) = 0. (8) In order to consider proper conservation of mass across the interfaces, the jump mass balance equations must be used. Note, that since equilibrium has been assumed at I-II interface, it is not necessary to write the jump mass balance for O 2. The jump mass balance for TiO 2 at I-II interface is written as: n (1) w (1) I O 2 II TiO 2 +O 2 Figure 2: Schematic of the I-II interface. N (II) n (1) c (II) w(1) n (1) = 0, (9) and, after dividing by c (II) it reduces to: The jump mass balance for each species at II-III interface is: TiO 2 : Ti : O 2 : N (II) N (III) ( v (II) w (1)) n (1) = 0. (10) n (2) c (II) w(2) n (2) = rσ M (TiO2 ) n (2) + c (III) N (II) n (2) = rσ M (O2 ) ; (11) w (2) n (2) = rσ M ; (12), (13) 5

6 n (2) w (2) II TiO 2 2 III Ti Figure 3: Schematic of the II-III interface. where r σ is the rate of production of given species due to the chemical reaction at the interface and M is the species molecular weight. The reaction at the interface (see assumption 1) can be written in terms of reaction rates as: r σ M = rσ M (O2 ) = rσ M (TiO2 ). (14) The above eq.(14) can be used to combine the jump mass balances (11)-(13) and to exclude the reaction rates. Thus, adding eq.(11) to eq.(13) one obtains: c (II) w(2) n (2) = ( c (II) v(ii) + N (II) ) n (2). (15) Similarly, subtracting eq.(12) from eq.(13) results: c (III) w (2) n (2) = ( c (III) v (III) + N (II) ) n (2). (16) Further, adding eqs.(11), (12) yields: ( v (II) w (2) γ(v (III) w (2) ) ) n (2) = 0, (17) where γ is the Pilling-Bedworth Ratio which is defined as: γ c(iii) c (II) = ρ(iii) ρ (II) M (TiO2 ) M. (18) It should be noted that only two of the three eqs.(15)-(17) are independent. One can make a list of the unknowns and available equations. Thus, there is a total of 10 unknowns: c (II) and three components for each of v (II), w(1) and w (2). However, 6

7 there are only 4 equations available: mass balance for O 2 in Region II (eq.(8)), necessary to compute c (II) ; mass balance for TiO 2 in Region II (eq.(5)), which gives an equation for determining v (II) ; and, two of the jump mass balances (eqs.(15)-(17)), which are used to compute the normal components of w (1) and w (2). Assuming that the tangential components of both w (1) and w (2) are zero, the number of unknowns is reduced by four. Finally there are four equations and six unknowns. Therefore, in order to solve the general 3D problem one must consider additional equations, i.e., momentum balance equations (see discussion in [23, p.532]). However, there are some cases when the oxidation problem can be solved without considering additional equations. These problems are one-dimensional (1D), such as planar oxidation, oxidation on the surface of a cylinder, oxidation of a cylindrical hole and oxidation of a sphere. It is mentioned here (see Appendix), that the model developed in this study is an extension of a class of existing models, capable of taking into account the volumetric expansion of TiO 2. However, the introduction of the volumetric expansion into the present model results in the need for the momentum balance equations, as mentioned above, which increases the complexity of the model. In the case of zero volumetric expansion, the general 3D oxidation problem can be solved without considering the momentum balance equations. 3 1D Oxidation As it was mentioned above, there are some special cases when one can solve the oxidation problem without solving the momentum balance. This section describes four different 1D oxidation problems - 1D planar oxidation, oxidation on the surface of a cylinder, oxidation of a cylindrical hole and oxidation of a sphere. All of these problems involve shapes of engineering significance, especially for Metal Matrix Composites (both fibrous and particulate) D Planar Oxidation In order to obtain a simple form of the governing equations, the frame of reference is chosen such that the I-II interface is stationary (see Fig.4). Then the mass balance for O 2 in Region II becomes: c (II) 2 c (II) (O D 2 ) (O2,TiO t 2 ) x 2 = 0, (19) with the following boundary conditions: c (II) = c 0 at x = 0, c (II) = 0 at x = h(t), (20) where c 0 is the equilibrium concentration of O 2 in titanium dioxide at the I-II interface and h(t) denotes the position of the metal-oxide interface. It is assumed that initially the oxide layer thickness is zero, i.e., h(0) = 0. Using eq.(15) the following condition for the x component of w (2) is obtained (since the problem is one-dimensional, y and z components 7

8 I II III TiO 2 w (2) Ti h(t) x Figure 4: Schematic of the planar oxidation problem. are equal to zero): w (2) x = D (O 2,TiO 2 ) c (II) c (II) x (21) x=h(t) The above model of planar oxidation is identical to the model of oxidation of silicon, derived in [14]. Let the following dimensionless variables be introduced c c(ii) c 0, x xˆl, t t D (O 2,TiO 2 ) ˆL 2, (22) where ˆL is the initial length of the specimen. Then eq.(19) can be written as and the interface condition in non-dimensional form becomes w x (2) = c 0 c (II) x x =h (t ) t 2 c = 0, (23) x 2 (24) where h (t ) is defined as h (t ) ĥ(t ) ˆL The solution of equation (23) is given by D (O2,TiO 2 ) ) = h( t ˆL2 ˆL. (25) c = A 1 + A 2 erf (η), (26) 8

9 where η is defined as η x 4t, (27) and the constants A 1 and A 2 are found from the boundary conditions: A 1 = 1, A 2 = 1 erf (λ). (28) In the equations above λ is the growth parameter and is defined as The interface condition (24) leads to λ h (t ) 4t. (29) λe λ2 erf(λ) = 1 π c 0 c (II), (30) which is a non-linear algebraic equation for the evaluation of λ. Substituting expressions for h (t ) and t into eq.(29), the following expression for the thickness of the oxide layer as a function of time is obtained: h(t) = λ 4tD (O2,TiO 2 ). (31) Once λ is known, eq.(31) can be used to compute h(t). On the other hand, if D (O2,TiO 2 ) is unknown, then one can use eq.(31) to estimate D (O2,TiO 2 ), using experimental oxide thickness measurements. 3.2 Oxidation in Cylindrical Coordinates The description of the oxidation problem in cylindrical coordinates is presented in this section. Two different problems are considered: oxidation on the surface of cylinder and oxidation of a cylindrical hole. For both of the problems the frame of reference is chosen such that Ti is not moving Oxidation on the surface of a cylinder An infinite axisymmetric solid cylinder exposed to an oxygen environment is considered (see Fig.5). R o and R i, shown in Fig.5 are positions of I-II interface (outer radius) and II- III interface (inner radius), respectively. Both R o and R i are functions of time and initially R o (0) = R i (0) = ˆR, where ˆR is the initial radius of the cylinder. Similarly as in section 3.1, the following non-dimensional variables are defined: c c(ii) c 0, r rˆr, t t D (O 2,TiO 2 ) ˆR 2. (32) Then, the governing equation in non-dimensional form becomes t 2 c r R c 0 2 r i (1 γ) r 9 c (III) r =R i c = 0, (33) r

10 I O 2 II TiO 2 R i III Ti R o O 2 Ti TiO 2 a) b) Figure 5: Oxidation on the surface of a cylinder: a) schematic of the problem; b) Scanning Electron Microscope (SEM) photograph of Ti wire oxidized for 72hrs at 700 C. with boundary and interface conditions where w (1) r c = 1 at r = R o, c = 0 at r = R i, (34) w (2) r = c 0 c (III) = (1 γ) R i R o R i = R i ˆR, r c 0 c (III) r =Ri r r =Ri, (35), (36) R o = R o ˆR. (37) There are available analytical solutions for mass diffusion in a cylinder with stationary boundaries (see, for example, [24]). Analytical solution for the moving interface problem is available only for the special case of oxidation from the center of a cylinder [9, p.111]. Therefore, to obtain the solution for the above described problem it is necessary to solve it numerically. The authors have implemented the Finite Differences Method, which has been recently used to solve 2D oxidation problems in metals for cases with zero volumetric expansion [11]. The formulation of the method starts with division of the Region II into N uniform intervals with a step r. A constant time step t is used throughout the computations. In establishing a finite difference scheme for eq.(33) the approach from [25] is used. In particular, the finite difference approximation of eq.(33) with backward-time scheme for discretization of the time derivative and forward-space scheme for the convective 10

11 term, which results in unconditionally stable scheme is (c ) k+1 j 1 r j 1/2 (c ) k j (c ) k+1 t 1 + R i (1 γ) j+1 2(c ) k+1 j + (c ) k+1 j 1 ( r ) 2 c 0 c (III) (c ) k+1 1 (c ) k+1 0 r (c ) k+1 j (c ) k+1 j 1 = 0, (38) r where the superscript k +1 indicates the number of the current time step while the subscript j indicates the current spatial position. Equation (38) can be written in matrix form as Ac k+1 = b, (39) with the matrix A being a function of the solution c k+1. An iterative procedure is used for solving (39). The solution of the previous iteration is used for evaluating A, i.e., c (s) k+1 is used to determine (s+1) c k+1, where s denotes the iteration number. The resulting system of linear equations is solved at each iteration using tridiagonal solver [26, p.414]. The iterative process is terminated when the norm of the difference of w r (2) between two iterations is less than After convergence is obtained, the positions of both I-II and II-III interfaces are updated using expressions given by (35) and (36). Different time and spatial steps were used to perform a convergence study of the numerical algorithm. Thus the number of intervals N was chosen to be equal to 100 and a time step of t = was used for numerical calculations. For these parameters a convergence is obtained after 2 3 iterations. The CPU time for the numerical code to run on a Pentium II-400 machine was approximately 2min. An alternative solution using a domain transformation has been implemented by the authors in an earlier work [13]. If the domain is transformed and the interfaces are immobilized, then the thickness of the oxide layer will enter into the governing equation, making it non-linear even for the case of γ = 1.0. Both the domain transformation method and the one used in this work use an iterative solution technique. For the problems considered here both methods converge after a few iterations. Comparison between the two numerical methods has been carried out for the 1D planar case. For this case the numerical results obtained by both methods converge to the analytical solution Oxidation of a cylindrical hole This section describes oxidation of a cylindrical hole embedded in an infinite medium with its surface exposed to an oxygen environment (see Fig.6). In this case R i indicates the position of I-II interface, while R o indicates the position of II-III interface. Similarly with the previous section 3.2.1, initially R o (0) = R i (0) = ˆR, where ˆR is the initial radius of the hole. Using the same non-dimensional variables as in the case of oxidation on the surface of a cylinder, the governing equation is written as t 2 c r 1 2 r 1 + R o(1 γ) 11 c 0 c (III) r r =R o c = 0. (40) r

12 III Ti II TiO 2 R i O 2 R o Ti O 2 TiO 2 a) b) Figure 6: Oxidation of a cylindrical hole: a) schematic of the problem; b) SEM photograph of a hole in Ti plate oxidized for 72hrs at 700 C. The boundary and interface conditions become w (1) r c = 0 at r = R o, c = 1 at r = R i, (41) w (2) = c 0 c (III) = (1 γ) R o R i r c 0 c (III) r =Ro r r =Ro, (42). (43) The same finite difference scheme and solution strategy as in the previous section are used to obtain the numerical solution to this problem. 3.3 Oxidation of a Sphere Another special 1D problem oxidation of a sphere is presented in this section. Consider a spherical particle exposed to an oxygen environment. The schematic of the problem is shown in Fig.7. Similarly with the previous section 3.2, the frame of reference is chosen such that v (III) = 0. R i and R o have the same meaning as in section and R o (0) = R i (0) = ˆR, where ˆR is the initial radius of the sphere. After taking into account the spherical symmetry, and using the same non-dimensional variables as in the previous two problems, the mass balance for O 2 in Region II in is written as: t 2 c r R 2 i 2 r r (1 γ) c 0 c (III) 12 r r =R i c = 0, (44) r

13 II TiO 2 R i III Ti R o I O 2 Figure 7: Schematic of the spherical oxidation problem. with boundary and interface conditions c = 1 at r = R o, c = 0 at r = R i, (45) w (2) = c 0 c (II) w (1) = (1 γ) R i R o r c 0 c (II) r =Ri r r =Ri, (46). (47) As in the case of oxidation of a cylinder, the analytical solution for oxidation in spherical coordinates is available only for the special case of oxidation from the center of the sphere [9, p.110]. Thus, the solution to eq.(44) is obtained numerically by means of the Finite Differences Method. The same finite difference scheme and solution strategy as in the previous two cases are used. 4 Results and Discussion 4.1 Estimation of the Diffusion Coefficient The experimental data used in the current study (taken from reference [27]) is for commercially pure Ti. The values of the densities of Ti and TiO 2 are taken to be 4500 kg/m 3 and 4250 kg/m 3, respectively, while the values of the molecular weights are 47.9 kg/kmole and 79.9 kg/kmole, respectively. Therefore, the values of the concentrations of Ti (c (III) ) and TiO 2 (c (II) ) are 93.9 kmole/m3 and 53.2 kmole/m 3, respectively and the value of γ is equal to Assuming that the values of c 0 for Ti and Ti-15-3 alloy are equal, the authors choose to use the value of c 0 for Ti-15-3 which has been experimentally measured and 13

14 reported to be equal to 12.5 kmole/m 3 [10]. Experimental data from [27] has been utilized to obtain the effective diffusivity coefficient D (O2,TiO 2 ). First, the value of λ is computed using eq.(30) and is equal to Then the value of D (O2,TiO 2 ) is computed using eq.(31) to obtain least-square fit of the experimental thickness measurements as shown in Fig.8. The estimated value of D (O2,TiO 2 ) for oxidation at 700 C is equal to µm 2 /s. It should be mentioned, that the computed diffusion coefficient may be different from its exact value since the value of c 0 used to estimate it is taken for a different alloy. However, if the exact value of c 0 is known, one can use the above described procedure to make a more accurate estimation of D (O2,TiO 2 ). 25 µm Figure 8: Experimental data for the oxide layer thickness as a function of time for pure Ti oxidized at 700 C. Also shown is the least-square fit of these data to estimate the value of D (O2,TiO 2 ). 4.2 Numerical Results Numerical results obtained for the different oxidation problems discussed above are presented in this section. One of the important features of the presented analysis which distinguishes it from the existing models is that it takes into account the difference in molar densities of TiO 2 and Ti, which results in expansion of the oxide during the oxidation process. For comparison reasons results presented in this section are for two different values of γ. The first is for the case of γ = 1.0 (no volumetric expansion) and the second one is for the case of γ = 1.766, which corresponds to the theoretical estimate based on density calculations. The computed oxide layer thickness as a function of time using the value of γ = 1.0 is shown on Fig.9 for three different problems, namely, 1D planar oxidation, oxidation on the surface of a cylinder and oxidation of a cylindrical hole. Figure 10 shows the same set of results for the value of γ = The maximum time of modeling of oxidation on the surface of the cylinder for both values of γ is chosen to allow the full oxidation of the specimen, while the maximum modeling time for the problem of oxidation of a cylindrical hole for the case of γ = is chosen to allow the full closure of the hole (i.e., R i = 0). 14

15 !#" $ h* = h R h* ( 2 2) t* = t D O,TiO R 2 γ = % t* Figure 9: Dimensionless thickness h of the oxide layer as a function of dimensionless time t for 1D planar oxidation problem, oxidation on the surface of a cylinder and oxidation of a cylindrical hole for the case of γ = # $ % h* = h R h* "! ( 2 2) t* = t D O,TiO R 2 γ = Figure 10: Dimensionless thickness h of the oxide layer as a function of dimensionless time t for 1D planar oxidation problem, oxidation on the surface of a cylinder and oxidation of a cylindrical hole for the case of γ =

16 It can be seen from Fig.9 that when γ = 1.0 the oxide layer grows faster for the case of oxidation on the surface of a cylinder than the oxide layer in the planar case, while the oxide layer for the case of oxidation of a cylindrical hole grows slower than the oxide layer in the planar case. This can be explained by considering that as the oxidation progresses the area of the reacting interface (II-III interface) decreases for the case of the oxidation on the surface of a cylinder, while it increases for the case of oxidation of a cylindrical hole, and it stays the same for the planar oxidation problem. However, for the case of γ = 1.766, it can be seen from Fig.10 that the oxide layer for both oxidation of a cylinder and oxidation of a hole grows faster than the oxide layer for the planar problem. Due to the volumetric expansion, the oxide in the case of oxidation of a cylindrical hole grows faster than the oxide layer in the planar oxidation case, dominating the effect of the increasing reaction surface area. In addition, one can observe a change in the character of the oxide layer growth for the case of oxidation of a cylindrical hole for γ = 1.766, i.e., on the late stages of oxidation the oxide starts growing much faster. This effect can be explained by the fact that at this stage the oxide layer growth is dominated by the expansion of the oxide since the radius of the hole is much smaller than the initial radius before oxidation. For the case of oxidation on the surface of a cylinder, the faster oxide layer growth at the late stages of oxidation shown in both Figs.9 and 10 is due to the reduced reaction surface area. Also, the time required for the full oxidation of the cylinder is increased for the case of γ = The numerical results for oxidation of a sphere are shown in Fig.11 for two different cases, the first one being the case of γ = 1.0 and the second one being the case of γ = Similarly as in the case of oxidation on the surface of a cylinder, the numerical procedure is run until the whole specimen is oxidized (i.e., Ri = 0). It can be seen from Fig.11 that at the early stages of oxidation the oxide layer grows slower for the case of γ = 1.0. However, when the radius of the sphere becomes much smaller than its initial radius, the growth of the oxide layer is significantly accelerated due the decreased reacting surface area (II-III interface). Another effect of the value of γ is the increased time required for the full oxidation of the specimen. The predicted oxide layer thicknesses for the cases of oxidation from the surface of a cylinder and oxidation of a hole are compared with the experimental data, obtained during experiments in the Materials and Structures Laboratory at Texas A&M University. The specimens used during experiments were made of commercially pure Ti (i.e., the same material used to estimate the diffusion coefficient). The comparison of the results for the case of oxidation from the surface of a Ti cylinder at 700 C for 72hrs. is shown in Fig.12 and the comparison of the results for the case of oxidation of a cylindrical hole in Ti plate at 700 C for 72hrs. is shown in Fig.13. For both cases, the value of D (O2,TiO 2 ) has been chosen from 1D planar oxidation data, as discussed in section 4.1. It can be seen from the last two figures, that the model underestimates the thickness of the oxide layer for the case of oxidation of a cylinder, while for the case of oxidation of a hole the thickness is overestimated. This effect can be explained after careful observation of the SEM photographs of oxidized specimens, shown in [27]. It is observed, that the oxide layer is porous and the porosity in the case of a 1D planar oxidation is higher than the porosity in the case of oxidation of a hole. However, it is lower than the porosity observed in the case of oxidation on the surface of a cylinder. Since the model can not account for porosity explicitely, the effect of porosity can implicitly be accounted for by choosing a modified diffusion coefficient for each case. 16

17 h* = h R h* ( 2 2) t* = t D O,TiO R γ=1.0 γ= t* Figure 11: Dimensionless thickness h of the oxide layer as a function of dimensionless time t for the case of oxidation of a Ti sphere at 700 C for two different values of γ. Another issue to be addressed is the experimentally observed fact that there is free oxygen in titanium [2]. Since the primary objective of this work is to investigate the effect of the expansion of the oxide layer during oxidation, the simplifying assumption 5 has been made. However, the current model can be modified to take into account the diffusion of oxygen into titanium. In this case, the assumption 1, stating that the reaction is instantaneous, must be modified to allow for the concentration of O 2 at II-III interface to be non-zero. The mass balance for O 2 must then be written for Region III, as well as the jump mass balance for O 2 at II-III interface must be modified to account for the presence of oxygen in Region III. 17

18 h* ! "$# %& '(#)* +,-.# /10, * /12 µm µm t* Figure 12: Non-dimensional oxide layer thickness h vs. non-dimensional time t for the case of oxidation from the surface of a cylinder h* 0.1!! "$# %'& () *+&,-. /01& 243 / µm µm 6 µm t* Figure 13: Non-dimensional oxide layer thickness h vs. non-dimensional time t for the case of oxidation of a cylindrical hole. 18

19 5 Conclusions This work represents a new approach to the solution of the problem of oxidation of metals in general and, specifically, of Ti. The experimental observed expansion of titanium dioxide during oxidation has been taken into account using the proper jump mass balance. The diffusion coefficient of O 2 into TiO 2 has been estimated using the simplified 1D model of planar oxidation and experimental data. Numerical solutions have been obtained for the problems of oxidation on the surface of a cylinder, oxidation of a cylindrical hole and oxidation of a sphere. Numerical results for the first two problems have been compared with the available experimental data. The model accurately simulate 1D planar oxidation experiments, while it underestimates the oxide layer thickness for the case of oxidation on the surface of a cylinder and overestimates the thickness for the case of oxidation of a hole. The observed discrepancy is attributable to different porosity of the oxide for different specimen geometries. The developed model is of a general nature and may be applicable to other metals. If experimental data is available, one can estimate the diffusion coefficient using the procedure described in section 4.1 and apply the developed model to predict oxide layer growth in nonplanar geometries. Oxygen dissolution in the metal can also be accounted for by a proper modification of the model. Extension to real 3D geometries will require introduction of the momentum balance equations which is a natural extension of the presented work. Acknowledgments The authors acknowledge the partial support provided by AFOSR grant No.F and express their appreciation for the experimental data provided by P.K.Imbrie and for his valuable comments and discussions. References [1] J. Stringer. The oxidation of titanium in oxygen at high temperatures. Acta Metallurgica, 8: , [2] A. E. Jenkins. J. Inst. Metals, 82: , [3] P. Kofstad, K. Hauffe, and H. Kjollesdal. Acta Chem. Scand., 12: , [4] J. E. L. Gomes and A. M. Huntz. Correlation between the oxidation mechanism of titanium under a pure oxygen atmosphere, morphology of the oxide scale, and diffusional phenomena. J. Metals, 14(3): , [5] P. Kofstad. High Temperature Oxidation of Metals. Wiley, NY, [6] C. Wagner. J. Metals, 4(91), [7] J. Unnam, R. N. Shenoy, and R. K. Clark. Oxidation of commercial purity titanium. Oxidation of Metals, 26(3/4): , [8] J. Crank. The Mathematics of Diffusion. Clarendon Press, Oxford, [9] J. Crank. Free and Moving Boundary Problems. Clarendon Press, Oxford,

20 [10] D. C. Lagoudas, X. Ma, D. A. Miller, and D. H. Allen. Modeling of oxidation in metal matrix composites. Int. J. Engng Sci., 33(15): , [11] P. B. Entchev, O. P. Iliev, and D. C. Lagoudas. Numerical simulation of a 2D oxide layer growth in an anisotropic medium. Journal of the Mechanical Behaviour of Materials, 7(1):67 84, [12] D. C. Lagoudas, P.B. Entchev, and R. Triharjanto. Modeling of oxidation and its effect on crack growth in titanium alloys. Journal of Computer Methods in Applied Mechanics and Engineering, In print. [13] Z. Ding and D. C. Lagoudas. A domain transformation technique in oxygen diffusion problem with moving oxidation fronts on unbounded domains. International Journal for Numerical Methods in Engineering, 42(2): , [14] K.-Y. Peng, L.-C. Wang, and J. C. Slattery. A new theory for silicon oxidation. J. Vac. Sci. Technol. B, 14(5): , [15] P. Murray and G. F. Carey. Compressibility effects in modeling thin silicon dioxide film growth. Journal of the Electrochemical Society, 136(9): , [16] N. B. Pilling and R. E. Bedworth. The oxidation of metals at high temperatures. J. Inst. Metals, XXIX(1): , [17] H. E. Evans. Stress effects in high temperature oxidation of metals. International Materials Reviews, 40(1):1 40, [18] N. Birks and G. H. Meier. Introduction to High Temperature Oxidation of Metals. Edward Arnold Ltd, 41 Bedford Square, London, WC1B 3DQ, [19] S. J. Bull. Modeling of residual stress in oxide scales. Oxidation of Metals, 49(1/2):1 17, [20] Chun Liu, Anne-Marie Huntz, and Jean-Lou Lebrun. Origin and development of residual stresses in the Ni-NiO system: in-situ studies at high temperature by X-ray diffraction. Material Science and Engineering A, 160: , [21] E. C. Aifantis. On the problem of diffusion in solids. Acta Mechanica, 37: , [22] Y. Weitsman. Stress assisted diffusion in elastic and viscoelastic materials. J. Mech. Phys. Solids, 35(1):73 93, [23] J. C. Slattery. Advanced Transport Phenomena. Cambridge University Press, [24] H. S. Carslaw and J. C. Jaeger. Conduction of Heat in Solids. Oxford University Press, New York, second edition, [25] A. A. Samarskii. Theory of Difference Schemes. Nauka, Moskow,

21 [26] R. L. Burden and B. J. Faires. Numerical Analysis. Brooks/Cole Publishing Company, 6 th edition, [27] P. K. Imbrie and D. C. Lagoudas. The morphological evolution of TiO 2 scale formed on various 1D and 2D geometries on titanium. Acta Metallurgica, Submitted. Appendix As it was mentioned, the developed model is an extension of a class of models reported in the literature (see, for example, [5, 9, 10]). To briefly show this, recall the model from [10]. Assuming that Ti is oxygen free, the governing equation is [10]: ĉ (II) t D (O2,TiO 2 ) ĉ (II) = 0, (48) where ĉ (II) is the total concentration of oxygen, i.e., free O 2 plus the immobilized oxygen contained in TiO 2. The boundary conditions are ĉ (II) = ĉ 0 at I-II interface ĉ (II) = [c] at II-III interface, (49) where [c] is defined as the jump of the oxygen concentration across the interface. After careful analysis of the model, one can conclude that the value of [c] shows the amount of immobilized (reacted) oxygen in TiO 2. The interface condition at II-III interface is w (2) n (2) = D (O 2,TiO 2 ) ĉ (II) ĉ (II) n (2). (50) Now it is easy to see the connection between the model from [10] and the developed model. First, the following equation relating ĉ (II) and c (II) holds: ĉ (II) = c (II) + [c]. (51) Then, taking into account the reaction stoichiometry (see assumption 1) it is seen that one mole of O 2 and one mole of Ti produce one mole of TiO 2. Thus, acknowledging the fact that c (II) = 0 at the interface, it is concluded that ĉ (II) = c (II) (= [c]). (52) Thus, it is possible to rewrite the above equations (48)-(50) in terms of c (II) and to obtain a form, similar to the one of the new model. However, there is one additional assumption which would make these two models equivalent, namely, the requirement that the concentrations of Ti and TiO 2 are equal. Then, in view of eq.(17), it follows that the velocities of both Ti and TiO 2 are equal, and, therefore, if the frame of reference is chosen such that either Ti or TiO 2 is not moving, then both v (III) the last term of eq.(8), involving v (II) will be identical to the model reported in reference [10]. and v (II) will be identically zero. Therefore, will be equal to zero. Thus, the developed model 21