Application of a Diffuse Interface Model (DIM) to the oxidation of W (Fe, Ni)-based alloys at high temperatures

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1 Academia Journal of Scientific Research 5(4): , April 2017 DOI: /ajsr ISSN: Academia Publishing Research Paper Application of a Diffuse Interface Model (DIM) to the oxidation of W (Fe, Ni)-based alloys at high temperatures Accepted Date 20 th, April 2017 ABSTRACT Abdulsalam Ahmed Alhazza Nanotechnology Program, Energy and Building Center, Kuwait Institute for Scientific Research, P.O. Box Safat- KISR- Kuwait. Corresponding author. ahazza@kisr.edu.kw The high-temperature oxidation of tungsten was achieved at different temperatures, covering oxidation rates from 750 to 1,000 C. A Diffuse Interface Model (DIM) was introduced to describe the high temperature internal oxidation. The zone has dissolved oxygen and metal atoms diffusing and reacting, thereby, resulting in an inward movement of the zone. The high-temperature oxidation data for tungsten were analyzed using a non-linear optimization method to obtain the optimal values for different parameters. The DIM was successfully applied for all temperatures in evaluating the velocity constant (k), the diffusion coefficients of the WO 3 product and core W and the gas film diffusion coefficient (k M). Keywords: Oxidation, diffuse interface model, optimization. INTRODUCTION In physical chemistry, transition metals exhibit outstanding properties. Tungsten oxide (WO 3) is an important material in electrical devices (Bose et al., 1989), catalysts (Butterworth et al., 1998) and chemical sensors (Cai et al., 1995; Dowding et al., 1994; Ekbom and Holmberg, 1992). The variation in electrical conductivity is one of the characteristic properties of WO 3, which can vary from a semi-conducting state (for WO 3) to a conducting state (for WO 2). Tungsten and the tungsten alloy group are used in a wide range of applications, for example, the coil of an incandescent lamp, the contact tip of an electrical switch, an automobile horn and even as components in nuclear fusion reactors or ion drive motors in space probes. The reason for this wide range of use lies in the several outstanding properties of tungsten such as its high melting point, low vapor pressure, high atomic number, good electrical and thermal conductivities (Jing-lian et al., 2008). In the W-O system, there are stoichiometric oxides, that is, WO 3, WO 2.9, WO 2.7, and WO 2, as well as nonstoichiometric structures that represent ordered or partially ordered defect structures of the oxygen-rich oxide in which the central W atom is octahedrally surrounded by six oxygen atoms. In WO 3, neighboring octahedral structures are in contact only at the corners which increases the oxygen deficiency (reduction and conversion to lower oxides) and common edges and surfaces are progressively formed (Yih and Wang, 1979; Alhazza, 2009). The aim of this work was to produce a homogeneous powder from heavy metal swarf for recycling. The process route was based on controlled oxidation to breakdown the swarf. First, the microstructure of the swarf was characterized using optical metallography, Scanning Electron Microscopy (SEM) and X-Ray Diffraction (XRD). This characterization was followed by oxidation to achieve mechanical breakdown and yield a friable oxide. In this step, TG was used to determine the temperatures for oxidation and provide insight regarding the kinetics of the process, while SEM and XRD were used to determine the morphology and type of oxide and the TG results used to select the temperatures for oxide manufacturing.

2 Weight loss g/cm 2 ) Academia Journal of Scientific Research; Alhazza C 800 C 850 C 900 C 950 C 1000 C Time (s) Figure 1: Experimental data for tungsten oxidation expressed as a parabolic rate. MATHEMATICAL MODEL The high-temperature corrosion rate of many common alloys can be described according to the three kinetic laws: linear, parabolic, or logarithmic (Khan et al., 1998). The rate of high-temperature corrosion of alloys follows the parabolic rate law, which requires that the square of the film thickness be proportional to time. If diffusion is controlled by the rate of gases or metal ions moving through a corrosion product layer (Khan et al., 1998), the rate of film growth follows a parabolic law which can be written as: w 2 = kt + ɛ (1) Where W is the weight gain of the specimen, k is known as the parabolic rate constant, t is the time, and є is a constant. The parabolic rate constant can be expressed (Khan et al., 1998) as: k=constant P 1/2 e (-E/RT) (2) The present data can be represented by Equation 2 (Figure 1) where the calculated activation energy is 80.4 kj/mole for tungsten corrosion in the temperature range of 750 to 1,000 C. The reaction (oxidation) between a metal or alloy and a gas in the absence of water or an aqueous phase is often called scaling, tarnishing, high-temperature corrosion, or dry corrosion (Khan et al., 1998). A diffused interface model (DIM) was introduced to describe the mechanism of high temperature oxidation for cobalt (Co), iron (Fe) and nickel (Ni). The high temperature oxidation data for these metals were analyzed using a non-linear optimization method to obtain the optimal values for the different parameters used in the DIM. The reaction zone dissolved oxygen and metal atoms diffusing into the zone and reacting and this result in the inward movement of the zone. The high temperature oxidation data for the metal-deficit (p-type) cobalt, iron and nickel were analyzed. The results of the model were successfully used to predict the experimentally determined parabolic rate constants for the oxidation of cobalt at high temperatures. The DIM is based on the rate constant of the reaction, the rate of diffusion of metal atoms in the oxide layer an d the rate of diffusion of an

3 Academia Journal of Scientific Research; Alhazza. 063 Figure 2: Starting material (Tungsten waste). oxidant in the corrosion product and in the un-reacted core for high temperature oxidation by oxygen at atmospheric pressure. The diffusion coefficients for the product layer and the unreacted metal core (cobalt, nickel or iron) were correlated with the parabolic rate constants and are represented by Equations 3 and 4 given as: D P= k g (3) V = (5) Where ρ is the density of tungsten in g/cm³. For the dimensions of the sample, the width is equal to X, the length is equal to 4X and the thickness is equal to 0.2X. Thus, the volume of the sample is 4X X 0.2X = 0.8X 3. Therefore, Equation 6 becomes: D C= K g (4) In this paper, the oxidation process rate for the tungsten alloy was parabolic above 750 C for all temperatures with alloying elements consisting of mainly Ni and Fe, which represent 4.6% of the alloy mass. The DIM was applied for the oxidation of tungsten at all temperatures used in the experimental work. The tungsten sample has a spiral shape (Figure 1) and is a long sheet of metal. The volume of the sample has three dimensions: length, width and thickness (Figure 2). With the mass (w) of the sample in grams and the volume (V) in cubic centimeters, the equation for the volume of the sample can be written as: 0.8X 3 = and X= (6) Khan et al. (1998) made a number of assumptions for the application of the DIM to the high temperature oxidation of metals. The assumptions are: (i) the scale is a homogeneous diffusion barrier, (ii) only a single oxidation product forms, (iii) the flux of oxygen is unidirectional and independent of the distance for a given scale thickness (quasi-steady state scale growth) and (iv) local equilibrium is maintained at the metal/scale and scale/gas interfaces throughout the thickness of the scale. These assumptions ensure that a parabolic oxidation growth is maintained. Consider the following irreversible gas/solid reaction:

4 Academia Journal of Scientific Research; Alhazza. 064 Figure 3: Schematic diagram for a small piece of tungsten. A(g) + bb(s) Products (7) A p-type or positive semi-conductor mechanism is due to either a deficit of metal or an excess of non-metal (intrinsic semi-conduction). This phenomenon is represented by the formation of a metal deficit p-type semi-conductor with cation vacancies and electron holes by the incorporation of oxygen into the metal lattice. Therefore, Equation 8 is given as: W + O 2 WO 3 (8) = b = = (11) (12) (13) The rate of reaction (r) can be written as: r = = -K q n (9) Based on the assumption made by Khan et al. (1998) that the diffusivity of the metal atom in the reaction zone is D M, the rate of diffusion was given as: The concentration gradient of metal ions can be interpreted as a function of the gaseous concentration. The oxygen concentration in the bulk gas can be assumed to be related to the diffusion through a gaseous film with a resistance equivalent to the gas-phase mass-transfer coefficient. Equating the gaseous rate of diffusion to the rate of diffusion into the product layer and the rate of diffusion in the reaction zone, the following relationship can be obtained between the bulk concentration of gas (C 0) and the interfacial concentration (C i) (Figure 3) is given as: = -D M (10) And the following Equations were derived (Khan et al., 1998): K M ( ) = D P = (14)

5 Academia Journal of Scientific Research; Alhazza. 065 Table 1: Example of the computer output used to predict the kinetic parameters for tungsten oxidation by air at 1,000 C. Number Experimental time (s) Predicted time (s) Weight loss (g/cm) Substituting C i in Equation 13, we obtain: = (16) (15) This equation can be integrated to obtain the calculated time (t calc) for the formed thickness of the corrosion product (L-X C): t calc = Where: (17) optimum values of the mass transfer diffusion constant (k M), the velocity constant of reaction (k), the diffusivity of oxidant gas in the core (D C), the diffusivity of the oxidant gas in the corrosion product (D P) and the fraction of the zone reacted ( ). The optimization technique is based on minimizing the following objective function: (22) Where is the experimental time for the i th observation. The experimental time is obtained from the corrosion rate data given for the oxidation reactions in Equation 8. The experimental fraction conversion (X) can be calculated from the corrosion rate data (Figure 3): (23) Where ƒ is the fraction of the zone that is already reacted. (18) (19) (20) RESULTS AND DISCUSSION High-temperature metallic corrosion is commonly expressed by a parabolic rate as given in Equation 1 (Table 1 and Figure 1) for the temperature range of 750 to 1,000 C. From the Arrhenius equation, the parabolic rate constant is related to the temperature (Figure 4). The magnitude of the activation energy for tungsten oxidation between 750 and 1,000 C, respectively is calculated as: Solution methodology (21) The solver function (Microsoft Office, 2007, London, UK) was used as a multi-variable search function to find the k =, (24) ( Where the slope is equal to E/R and the gas constant (25)

6 Parabolic rate constant Academia Journal of Scientific Research; Alhazza. 066 Figure 4: Parabolic rate constant for tungsten oxidation. Figure 5: Arrhenius-type plot for the diffusion coefficients of oxygen for tungsten corrosion. R=8.314 kj/mol. The activation energy for the parabolic rate constant is E= =80.48 kj/mol. DIM was applied to the corrosion data at different temperatures and various parameters evaluated using the built-in solver function in Microsoft Excel. The most important parameters are the product diffusivity (D P), core diffusivity (D C), rate constant (k) and gas film mass transfer coefficient (k m). Parameters such as k and k m have high values and are nearly constant for all temperatures, whereas D P and D C are related to temperature by the Arrhenius equation. Figure 5 shows the dependency of D P and D C on temperature.

7 Diffusion coefficient (cm 2 /s) Academia Journal of Scientific Research; Alhazza. 067 Parabolic rate constant (kg 2 /cm/ min) Figure 6: Diffusion coefficients in the core and product layers versus the parabolic rate constant for tungsten. Figure 6 is a plot of the experimentally obtained parabolic rate constant versus the diffusion coefficients in the reaction zone predicted by the DIM. The core and product layer diffusion coefficients (cm 2 /s) are related to the parabolic rate constant k g (gm 2 /cm 2 -s) according to the Equations 26 and 27: D P=24.6 k g (26) D C=20.91 k g (27) This equation is valid for tungsten at temperatures of 750 to 1,000 C. Thus, the DIM was able to predict a universal relationship between the experimentally determined parabolic rate constant and the model-predicted diffusion coefficients of oxygen in the product layer and reaction zone. The effect of temperature is simply related to the value of k g since the experimental values of k g are determined by plotting the weight gain squared against time for different temperatures. Conclusion Experimental data for the high-temperature oxidation of tungsten were described by a parabolic rate expression. The parabolic rate constants were related to temperature by the Arrhenius equation, where the activation energy equaled 80.4 kj/mol. The DIM was applied successfully for all temperatures in evaluating the velocity constant (k), the diffusion coefficients of the WO 3 product and core W and the gas film diffusion coefficient (k M). The values of the rate constant k and gas film resistance were large and not dependent on the temperature. The diffusion coefficients of the product and core were functions of temperature and related by the Arrhenius expression, resulting in the same activation energy. These diffusion coefficients were also related to the parabolic rate constant, allowing one to predict the value of the core diffusivity or the product diffusivity from the parabolic rate constant. REFERENCES Alhazza AA (2009). Oxidation and reduction of tungsten alloy swarf. Int. J. Refract. Met. Hard Mater. 27 : Bose A, Jerman G, German RM (1989). Rhenium alloying of tungsten heavy alloys. Powder Metall. Int. 21 :9-13. Cai WD, Li Y, Dowding RJ, Mohamed FA, Lavernia EJ (1995). A review of tungsten-based alloys as kinetic energy penetrator materials. Rev. Particulate Mat. 3: Dowding RJ, Hogwood MC, Wong L, Woodward RL (1994). Tungsten alloy properties relevant to kinetic energy penetrator performance, in Proceedings of the Second International Conference on Tungsten and Refractory Metals (Metal Powder Industries Federation, Princeton, NJ. p.3.ekbom L, Holmberg L, Persson A (1992). A tungsten heavy alloy projectile with spiculating core. Tungsten & Tungsten Alloys 1 :

8 Academia Journal of Scientific Research; Alhazza. 068 Butterworth GJ, Forty CBA, Turner AD, Junkison AJ(1998). Recycling of copper used in fusion power plants. Fusion Eng. Des. 38 : Jing-lian F, Tao L, Hui-chao C, Deng-long W (2008). Preparation of fine grain tungsten heavy alloy with high properties by mechanical alloying and yttrium oxide addition. J. Mater. Process. Technol. 208: (2008). Khan AR, Alhajji JN, Reda MR (1998). Generalized diffuse interface model for determination of kinetic parameters in high-temperature internal corrosion reactions (gas/solid system). Int. J. Chem. Kinet. 30: Yih SWH, Wang CT (1979). Tungsten: Sources, Metallurgy, Properties and Application Plenum Press, New York.. Cite this article as: Alhazza A (2017). Application of a Diffuse Interface Model (DIM) to the oxidation of W (Fe, Ni)-based alloys at high temperatures. Acad. J. Sci. Res. 5(4): Submit your manuscript at: