Oxidation Kinetics of Aluminum Powders in a Gas Fluidized Bed Reactor in the Potential Application of Surge Arresting Materials

In this technical paper, the oxidation mechanism and kinetics of aluminum powders are discussed in great details. The potential applications of spherical aluminum powders after oxidation to be part of the surging arresting materials are discussed. Theoretical calculations of oxidation of spherical aluminum powders in a typical gas fluidization bed are demonstrated. Computer software written by the author is used to carry out the basic calculations of important parameters of a gas fluidization bed at different temperatures. A mathematical model of the dynamic system in a gas fluidization bed is developed and the analytical solution is obtained. The mathematical model can be used to estimate aluminum oxide thickness at a defined temperature. The mathematical model created in this study is evaluated and confirmed consistently with the experimental results on a gas fluidization bed. Detail technical discussion of the oxidation mechanism of aluminum is carried out. The mathematical deviations of the mathematical modeling have demonstrated in great details. This mathematical model developed in this study and validated with experimental results can bring a great value for the quantitative analysis of a gas fluidization bed in general from a theoretical point of view. It can be applied for the oxidation not only for aluminum spherical powders, but also for other spherical metal powders. The mathematical model developed can further enhance the applications of gas fluidization technology. In addition to the development of mathematical modeling of a gas fluidization bed reactor, the formation of oxide film through diffusion on both planar and spherical aluminum surfaces is analyzed through a thorough mathematical deviation using diffusion theory and Laplace transformation. The dominant defects and their impact to oxidation of aluminum are also discussed in detail. The well-controlled oxidation film on spherical metal powders such as aluminum and other metal spherical powders can potentially become an important part How to cite this paper: Shih, H. (2019) Oxidation Kinetics of Aluminum Powders in a Gas Fluidized Bed Reactor in the Potential Application of Surge Arresting Materials. Materials Sciences and Applications, 10, 253-292. https://doi.org/10.4236/msa.2019.103021 Received: January 15, 2019 Accepted: March 26, 2019 Published: March 29, 2019 Copyright © 2019 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access


Introduction
Oxidation of aluminum to form aluminum oxide film has been playing significant roles in modern industry and receiving wide applications. Aluminum hard anodization has been widely using in semiconductor industry as chamber surface coating. The anodized aluminum through different oxidizing processes at low temperature in acidic solutions has demonstrated excellent performance in corrosion resistance for semiconductor plasma etching equipment from 200 mm silicon wafer fabrication to current 300 mm silicon wafer fabrication worldwide [1]- [28]. The high corrosion resistance, production repeatability, and low cost of anodized aluminum to form a unique oxide layer on aluminum alloys in acidic solutions have been widely used in semiconductor IC industry. Shih summarized the chamber materials performance for plasma etching chamber including anodized aluminum in his article [1]. Many technical papers have been published in the study of anodized aluminum and its applications [1]- [28]. The formation of oxide layer through logarithmic, parabolic, linear, and combinations of reaction rate in gas phases and the reaction mechanism of aluminum oxide formation described by many authors [29]- [42]. Aluminum powders can form a very thin oxide film in gas phases. Aluminum powders with a very thin oxide film may serve as switch material or called surge arresting materials (SAMs) when the aluminum powders with thin oxide layer and filler mix together. SAMs are electronic composite materials consisting of insulatively coated conductive and semiconductive particles which are embedded in a polymer binder or ceramic matrix. The calculation of ratio of mixing of aluminum powders and filler is based on the percolation theory [43] [44] [45] [46] [47]. From the percolation theory and the so-called "quantum mechanical tunneling", SAMs can serve as switches which can pass voltage or current at a sufficiently high electric field. It means that the oxide film on aluminum powders can serve as a controller to switch on or switch off at a critical voltage or current. This application may bring a great interest for people to create a passive switch material in SAMs study.
As an n-type of semiconductor, aluminum oxide (Al 2 O 3 ), is a metal excess oxide. The oxidation mechanism and the formation of metal oxide on metal surface at high temperature are summarized by Jones in his book [29]. Alumi [30]. According to PBR, the oxide will fail to cover the entire metal surface and will be non-protective when PBR is less than 1.0. The oxide will be protective when PBR is larger than 1.0. But PBR principally has a very limited application [31] and is only one of many factors which can determine the protective properties of an oxide film on metal. In the case of aluminum, the density of Al 2 O 3 is less than aluminum metal, the oxide layer can cover the entire surface of aluminum, and the aluminum oxide is very protective. Typical PRB and the properties of metal oxides are summarized in Table 1 [29] as shown below.
Ellingham diagram has been used to show the relationship of Gibbs free energy versus temperature for oxidation of metals [32] [33] [34].
In order to make the oxide film on metals as switch material, one has to control the growth rate and the uniformity of the oxide film on metals very well. It means that the thickness of oxide on metals and the uniformity of thickness of oxide layer have to be controlled very well during oxidation. The major properties are the thickness of oxide layer and the uniformity of the oxide film. Taking consideration of aluminum powder oxidation, the selection of aluminum powders and the methodology to form the aluminum oxide film are very important. 1) Excellent temperature control to reach isothermal conditions.

4) Catalytic gas reactions.
5) Aeratable powders have the most desirable fluidization properties. 6) Interparticle forces in the particles are significantly smaller than hydrodynamic forces and gas bubbles are limited in size.
The heat transfer, minimum fluidization velocity, fluidization technology, and packing properties have been discussed by various authors [49]- [54]. In order to operate the fluidized bed in bubble conditions and to avoid the occurrence of slugging, channeling, jetting, and spouting, the experimental design is very important. Many factors are involved in this technique. The bubble formation and types of fluidization are shown in Figure 1 and Figure 2, respectively.
Although gas fluidization bed has been using in different applications for many years, there is no theoretical modeling available. It means that the oxide formation relies on experimental results. The author needs to develop the theoretical model based on the dynamic system with specific hydrodynamic flow conditions. The current work is to create and to build up a theoretical mathematic modeling based on the real dynamic system and system parameters to estimate the thickness of oxide layer in a gas fluidization bed. Therefore, one can carry on the theoretical calculation and compare the results with experiments. In fact, the mathematical model developed in this work can help to carry out quantitative calculation of oxide growth rate at different conditions and compare the calculation with experimental results. Taking aluminum ball powder as an example, the author successfully created and developed the mathematical modeling and applied the model to compare with experimental results on a gas fluidization bed [55]- [60].

Theoretical Calculation of a Gas Fluidization Bed Reactor (FBR)
1) Basic parameters used in the kinetic calculation of a gas fluidization bed reactor: π = 3.1416.
S sur = πD 2 = surface area of aluminum ball with diameter, D. 3) Basic calculation of a gas fluidized bed reactor In order to carry out the calculation of the fluidization bed and the temperature-dependent relations of parameters in fluidization bed reactor (FBR), Shih wrote a general computer program 1 called "FLOW.EXE". This computer program was used to carry on the complete calculation of all important parameters in a FBR. It generates a very large database which can be used as a handbook of a fixed FBR. It basically carries on nine calculations. The particle size is between 10 μm and 100 μm, and the FBR temperatures are between 25˚C and 480˚C in the calculation. In order to reduce the generated database, a software program 2 called "FLOW1.EXE" was written by Shih [55] [56] [57] [59] [60] for FBR calculation at a fixed average particle size and at a user defined range of temperature.
Details of calculations of "FLOW.EXE" and FLOW1.EXE" are listed below: a) Average particle size vs. number of particles. b) Average particle size vs. effective surface area. e) The height of the gentle settled bed.
f) Relationship between diameter of aluminum powder and H max . g) Relations of particle size, Archimedes number, and minimum fluidization velocity. h) Thermal Conductivity of Gases at Different Temperatures. 1) Diameter, particle number, and surface area of spherical aluminum powders The relations of average particle diameter, particle number, and effective surface area of aluminum powders are calculated and shown in Table 2 and plotted in Figure 3 and Figure 4, respectively. In the calculation, the total volume of aluminum powders in taken as 1.0 m 3 . The average spherical aluminum particle size is between 10 μm and 100 μm. The effective surface area is in m 2 . With the increase of average particle diameter (d v ), both particle number (N p ) and effective surface area (S p ) decrease (see the table of nomenclature for the definitions of parameters).    2) The density and specific heat of air at various temperatures The density and specific heat of air at different temperature is calculated and summarized by Shih [59] [60] in Table 3 and plotted in Figure 5 and Figure 6. The specific heat of air reaches the minimum at about 273 K.
3) Viscosity of air, argon, and oxygen at various temperatures The viscosity of air, argon, and oxygen at different temperatures is calculated and summarized in Table 4 and plotted in Figure 7 by Shih [59] [60]. The viscosity of air is less than that of oxygen and argon. 4) Calculation of heat transfer One of the principal attractions of fluidized bed is the high rate of heat removal and addition. There are three modes of heat transfer which are of interest in gas fluidized bed.
• Gas to particle.
• Particle to particle.
• Particle to heat transfer surface.
For particles less than 1mm diameter, the overall heat transfer rate between a fluidized bed and gas entering through the distributor plater is so large that the gas attains the bed temperature within a few centimeters. For particles less than 100 μm diameter, the overall heat transfer rate between a fluidized bed and gas entering through the distributor plate is even larger that the gas attains the bed temperature within 1 centimeter. Since the high thermal conductivity of aluminum powders, the heat transfer among aluminum particles can be negligible. In fact, aluminum powders are pre-heated in argon environment and reach the temperature of the fluidization bed. Therefore, the surface-to-bed heat transfer coefficient can reach a maximum value α max and it can maintain over a wide range of velocity above the U mf . By considering the masked surface by bubbles, the Zabrodsky correction for α max is shown below: The overall heat transfer coefficient, α o , is shown as   Assuming the heat height of aluminum powder is about 22.0 centimeter, the total heat through the wall to the fluid of the bed can be calculated as For a given fluidized bed from 25˚C to 200˚C as an example, the total heat is Assuming that 1.5 pounds of aluminum powders have been pre-heated in argon environment to 200˚C, the total heat to raise air from 25˚C to 200˚C as an example at a constant flow rate is shown below: Since the volume of flow rate of air is fixed as From Equation (14), the calculated air velocity equals to the actual air velocity.
The conclusion is that for particles with diameter less than 100 μm, air can attain the bed temperature as soon as it enters the bed. By increasing the flow rate by 100 times, A air increases 100 times and the heat length equals 2.2 centimeters. Hair is function of D in , T, d p , V, and J air .

5) Maximum bed height
In order to keep the fluidized bed in bubble condition without slugging, the maximum bed height (H max ) and the inner diameter of the bed should satisfy the following relation: For a given fluidized bed with 0.070 m inner diameter, the maximum allowable depth for 100 μm to 10 μm diameter aluminum powders, the maximum bed depth should be less than 19.7 cm and 39.3 cm, respectively (see Table 5). The maximum allowable depth of the bed decreases when the diameter of aluminum particles increases. From Equation (19)  The relation will be used for the comparison.
The minimum heated length from Equation (21) is 22.0 cm and the maximum bed depth should be less or close to L to avoid slugging.

6) The Maximum bubble diameter
The maximum bubble diameter at the surface of a bed can be calculated as Assuming the free fall velocity of the particles equals 0.1 m/sec., the maximum bubble diameter is shown below: 7) The Average particle size The average particle size can be calculated by Equation (24) ( ) d p is taken as 20 μm in the calculation.

8) The Height of the gently settled bed
The gently settled bed is calculated as follows: The basic equation to calculate Archimedes number, A r , is as below: ( ) and the minimum fluidization velocity is For particles with 20 μm diameter and at 200˚C, A r is given as and 0.10 cm sec There are many factors which effect Archimedes number and minimum fluidization velocity. Table 6 and Table 7 show the relations among particle size, temperature, Archimedes number, and minimum fluidization velocity. Basically, at certain temperature with d v increasing, both A r and U mf increase. For the same particle size with temperature increasing, both d v and U mf decrease. Table 6 and Table 7  10) Thermal conductivity of air, argon, and oxygen  ( ) Figure 8. Relation between particle number and archimedes number. The thermal conductivity of air, argon, and oxygen is calculated in Table 8 and plotted in Figure 10 by Shih [59] [60]. The thermal conductivity of air and oxygen is very similar, but the conductivity of argon is much lower than the values of oxygen and air.

Model of Velocity and Pressure Distribution of a Gas Fluidization Bed Reactor
After the basic calculation, we can start to develop the dynamic model of velocity of pressure in the defined gas fluidization bed. The geometry of the fluidized bed is shown in Figure 11. The bed height is H in meter and the gas flow rate is Q in m 3 /s. There are n holes on the gas distributor, the radius of the hole is R. Assuming the particles are gently settled and when the bed is fluidized, the gas flow from the distributor forms n flow pipes through the bed with radius of the pipe equals to R. We also assume that the flow is evenly distributed among the n pipes. Therefore, the flow rate in each pipe is From Equation (12), one has Since R e is very small, the flow is laminar flow.

2) Governing Equations and Boundary Conditions
We chose cylindrical and Spherical coordinate system as illustrated in Figure   12. The coordinate center is at each flow pipe's centerline on the distributor surface (Z 1 ). Assume the flow is steady and incompressible, i.e., change of gas density is negligible, the basic equations for gas flow in the cylindrical and spherical coordinates [61] [62] are as follows: Continuity: R-direction momentum equation: Equation (41) is the Laplace operator of V r . θ-direction momentum equation: z-direction momentum equation: ∇ 2 V θ and ∇ 2 V z are the Laplace operators of V θ and V z , respectively.
Since the radius of the pipe is very small and the bed height is much larger than the pipe's radius, it is reasonable to say that the dominant flow velocity component is along Z direction. The rest of two components is negligible. The flow in the pipe is also axisymmetric. There two conditions can be written as Substitute Equations (44) and (45) where g is the gravity.
Substitute Equations (51)-(53) into Equations (47)-(49), one obtains From Equations (54) and (55), we know that P is only a function of Z, therefore, Equation (56) can be written as The boundary condition at the pipe wall is

3) Solution of the Equations
Integration of Equation (57), on obtains where C 1 is an integration constant and the density ρ is assumed to be constant.
Integrate Equation (59) where C 2 is another integration constant. At r = 0 (pipe axis), V z is finite, therefore, using Equation (59) Therefore, one has In the pipe, maximum velocity occurs at r = 0 To calculate the mean velocity, we need to calculate flow rate q from the V z distribution.
Mean velocity is shown below: It is interesting to note that ,mean ,maximum Equation (65) can be rewritten as Integrate Equation (67), we then have where P 1 is the pressure at the inlet of the pipe Z 1 .
We assume the variation of gas density inside the bed is negligible as discussed before, i.e., constant ρ = Thus, Equation (68) becomes From Equation (70), one can see that the pressure of air depends on many factors. The factors are T, μ, ρ, Z, q, R, and V z,mean . At a given temperature and for a fixed bed, T, μ, ρ, Z, q, and R are constants, the pressure difference is only a function of Z. The concentration of oxygen is proportional to the pressure at a Z position. Therefore, the bulk concentration of oxygen at different Z positions can be estimated using Equation (70).

1) Calculation of air pressure in FBR.
We have the following relation as shown in Equation (70). id density ρ can be calculated and obtained from Program "FLOW.EXE" and FLOW1.EXE, so as to V z,mean (V ave ) and H max , P can be calculated using Equation (71).
Since ρ and μ are function of temperature, therefore, P is function of μ, ρ, V z,mean , Z, T, and R and R is the radius of flow pipe. Z is also a function of particle size. We can express these relations as follows (see Equation (72)): where Q is the overall flow rate entering the FBR, T is the temperature, and Comp is the composition of gas phase in FBR.
From Equation (72), one can see that P decreases linearly with Z increasing and the typical calculation is shown below when the following parameters are selected in the calculation: 0 C 20 T =˚; In this case, the concentration of oxygen is uniformly distributed in the FBR. In this case, the concentration of oxygen is uniformly distributed in the FBR.
From the above calculation, one can get the following conclusions: -Oxygen is uniformly distributed in a FBR with large size of particles.
-Oxygen is not uniformly distributed in a FBR with small size of particles.
-The effect of gravity to the pressure P is negligible.

2) Estimation of Diffusion Coefficient in FBR
From the theory of random walk analysis with a vacancy mechanism, one has where D o is specific diffusion coefficient in cm 2 /s; n is the nearest position of jumping; ω is the frequency to jump from a position to a specific nearest neighbor and is about 10 12 to 10 13 s −1 ; α is the jumping distance of lattice sites. As D. D. Macdonald proposed a "dry" oxidation of aluminum [63], aluminum oxide grows as bilayer structure. The inner layer is due to the movement of oxygen vacancies from metal/film interface and the outer layer is due to the movement of cations outward from film/environment interface. Only barrier layer is considered to contribute to passivity.
The overall reaction of the oxidation is shown as below: The principal crystallographic defects are 1) Vacanies: Considering the cation vacancy The effect of oxygen pressure to the dominant oxide defect can be studied by using Equations (89) and (91).
Let's first consider the effect of oxygen pressure to oxygen anion vacancy at the worst case when aluminum powders have an average diameter as of 10 μm (see Equation (73)). From the above calculation, we can conclude that oxygen pressure in FBR at current working condition has no effect to the concentration of oxygen anion vacancies. The maximum variation of oxygen anion vacancies is below 5% for aluminum spherical powders with at the 10 μm diameter as the worst case 1. The vacancy concentration increases about 5.0% at Z max (H max ) position (0.39 m).
For aluminum powder with 50 μm diameter or larger, oxygen pressure has no effect to both oxygen anion vacancies and cation vacancies. The maximum variation on defect concentration of 50 μm diameter aluminum powder, for example at 200˚C for both cases, is less than 0.4% (see Case 2 and Case 3 for Equations (74) and (75)). where ξ is the thickness of the oxide, D o is the diffusion coefficient of oxygen, and r is a constant.

4) Oxidation on A Planar Aluminum Surface
From Fick's 2 nd law, the diffusion of oxygen and metal is written as where D Al is the diffusion coefficient of aluminum species, N o and N Al are atomic concentrations of oxygen and aluminum species, respectively.
In order to solve the differential equations, the following boundary conditions are assumed: The thickness of the oxide film after 60 minutes oxidation from Equation (121) is shown in Table 9.

5) Oxidation on Spherical Aluminum Powders
General diffusion in sphere has been studied by Crank [64], Shewmon [65], Galus [66], and Bitler [67] for different applications and for the FBR oxidation of aluminum powders as well as the surge arresting materials by Shih [ [72].
The growth rate of oxide layer can be estimated from the total amount of diffusing substance entering (oxygen lattice anions) or leaving (aluminum ions) the sphere in a gas fluidization bed considering the initial oxide thickness at time zero. Typical calculation of oxide thickness using software "APRIL.EXE" [59] [60] [70] and the comparison between experimental thickness in the FBR and the theoretical calculation based on the dynamic model is shown the the attached tables as below.
In Table 10 Table 10, the oxide layer thickness through theoretical calculation using the model developed can meet experimental oxide layer thickness in a fluidization bed between 473 K and 573 K for aluminum powders with a 10 μm diameter.
In Table 11, the relation of M t,exp and M t,cal for Oxygen Anion Vacancy Mechanism is calculated using computer software "APRIL.EXE" at various temperatures when r o = 60 μm, f(r o ) oxy = 1.0, and M init = 0.000125 (25 Å) are considered.
From Table 11, the oxide layer thickness through theoretical calculation using the model developed can meet experimental oxide layer thickness in a fluidization bed between 473 K and 573 K for aluminum powders with a 60 μm diameter.
In Table 12, the relation of M t,exp and M t,cal for Oxygen Anion Vacancy Mechanism is calculated using computer software "APRIL.EXE" at 373˚ K when r o = 10 μm, f(r o ) oxy = 1.025, and M init = 0.00075 (25 Å) are considered.
From Table 12, the oxide layer thickness through theoretical calculation using the model developed cannot meet experimental oxide layer thickness with optimized temperature when the oxidizing temperature in the calculation is only taken at 373 K for aluminum powders with a 10 µm diameter.
In Table 13, the relation of M t,exp and M t,cal for Oxygen Anion Vacancy Table 11. Relation of M t,exp and M t,cal for oxygen anion vacancy mechanism assuming

Summary and Conclusions
1) The mathematical modeling considering mass, momentum, energy conservation, effects of powder size, temperature, gas velocity, and other related factors has been developed for aluminum powder oxidation in a gas fluidized bed. This is the 1 st modeling for a quantitative calculation of thickness of aluminum oxide in a gas fluidization bed.
-At cross section of the pipe, the distribution of V z is parabolic and V z does not change along Z direction. -Pressure is linearly decreasing along Z direction and it is proportional to the pressure of air.
2) In the mathematical model, Equations (65) and (70)  3) The oxidation kinetics of planar and spherical aluminum powders has been discussed in great details. A uniform temperature field and a nice distribution of gas phase provide the uniform oxide growth rate of aluminum powders in the gas fluidization bed. 4) The basic theories of oxidation and growth rates have been summarized. The most possible growth rate of aluminum powders at low temperature initially is linear due to the surface boundary process or reaction is rate-determining. After initial oxidation the growth rate will follow parabolic law due to the diffusion processes in the growth of aluminum oxide. In this case, Wagner's mechanism holds.
5) The oxidation mechanisms of aluminum powders can be determined by interstitial cations diffusion process or anion vacancies diffusion process. As an n-type semiconductor, the rate-determining process of aluminum oxide is diffusion.
6) If the oxidation is controlled by a diffusion process, the growth rate of the oxide layer can be calculated by solving partial differential equations. Mathematical models have been developed and presented in the article. The modeling of internal oxidation process has also been discussed.
7) The growth rate from the experimental results is compared with the estimation of the models developed in this article. Very good correlation between theoretical calculation and experimental results are observed.
8) The optimization of gas fluidization of the oxide of aluminum powders involve many factors which have to be considered in the theoretical analysis of mathematical modeling. The detail information can be obtained through theoretical calculation and its comparison with experimental results. A computer program written by the author has been used in the calculation. An example calculation has been shown in the technical paper at a fixed temperature, particle size, gas velocity, bulk concentration of oxygen and aluminum, Archimedes number, minimum fluidization velocity, bed depth, and effective surface area.
9) The oxidation at high temperature generates stress in the outer oxide films on aluminum particles and leads to the formation of cracks. With the increase in the intensity of crack formation, the oxidation rate increases markedly, particularly for the γ-α-Al 2 O 3 transition. The oxidation film is not protective at high temperature oxidation. By controlling the oxidation at low temperature with the variations of oxygen concentration and relative humidity, the oxide layer is more protective and uniform. }