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Abstract

In this article, kinetics equation of the metal surface oxidation is presented using the perturbation approach. A moving boundary is converted into a fixed one by Landau transformation. The perturbation results show that (1) the solution of the zeroth-order stands for the quasi-steady state and that of the first-order is instantaneous and (2) the reaction rate, diffusion coefficient, and the thickness of the initial oxide layer have important effects on the oxidation kinetics. If the chemical reaction is supposed to be instantaneous, our results are reduced to the classical parabolic law, no matter whether there exists the initial oxide layer or not. Moreover, the kinetic coefficient of the parabolic law is analytically presented. The proposed perturbation scheme can be easily extended to the higher order, and the higher order solution will provide a method to correct the error offered by the classical model.

Keywords

Metal surface oxidation perturbation approach oxide kinetics Landau transformation

Introduction

Metals or complex alloys are widely used in the industry as a construction material.^1–6 When they are exposed to an oxidizing environment at high temperature, their surface will be gradually oxidized, and this leads to the degradation of the composites through the formation of a brittle oxide layer on the metal surface. If the produced oxides can form a relatively thick and adherent layer over the metal surface, metals or alloys will be protected by oxides. On the contrary, metals or alloys will proceed to be further oxidized. Thus, the extent of erosion will be accelerated, and the lifetime of metals or alloys will be shortened. The lifetime of metals or alloys is affected by many factors, and the oxidation is one of the most important factors. As well known, oxidation strongly depends on the velocity of the oxidation erosion, that is, the oxidation kinetics. Therefore, it is essential to investigate the oxidation kinetics of the metals.

Wagner⁷ first originated and developed the important theoretical formulation about the oxidation kinetics. Subsequently, much progress has been made in investigating the oxidation of metals. At present, the existing work is mainly experimental. However, the experimental kinetic coefficient “k_p” is valued on the basis of instantaneous reaction and diffusion. Kim and Moorhead,⁸ Bellosi et al.,⁹ and Osborne and Norton¹⁰ found that the chemical reaction or diffusion controls the oxidation and the kinetics of the oxidation satisfies the well-known linear rate, parabolic rate, and so on. In fact, the kinetic of the oxidation is also related to the temperature, oxygen partial pressure, volume, and the initial thickness of the oxide scale. Recently, Chou and colleagues^11,12 proposed a new kinetic model to describe the oxidation reaction under the all kinds of possible conditions and analytically expressed the reacted fraction as a function of oxidation time, temperature, and oxygen partial pressure. However, Chou only considered the oxidation kinetics controlled by the chemical reaction.

In addition, most research assumed that oxygen diffuses through the oxide and then reacts with the metal at the oxide/metal interface, that is, internal oxidation.^13–15 In the internal oxidation, some researchers thought that no oxygen dissolves into metal and others considered oxygen diffusion into the metal.^1,2,16 However, in the practical engineering, the oxidation process is sometimes controlled by metal ion diffusion. Namely, metal ion diffuses through the oxide layer and arrives at the oxygen/oxide surface and then reacts with oxygen and further forms a fresh oxide. For example, nickel will be oxidized at the surface when the temperature is higher than 1370 K and an oxygen partial pressure is between 0.2 and 1.¹⁷ In addition, zinc ions will diffuse in ZnO.^18,19 Therefore, metal surface oxidation should be paid more attention.

One of the important characters during the oxidation is nonlinear moving boundary, as discussed in Hu and Shen.²⁰ However, linear fixed boundary^21–25 is usually considered for the sake of the simplification. During the process of the oxidation, volumetric expansion or contraction occurs because the density of the oxide is typically less or larger than that of the metal. 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.²⁶ If the value of PBR is greater than one, there is volume expansion, while if the value of PBR is less than one, the oxide shrinks. Due to this volumetric expansion or contraction, and the growth of the oxidation scale, their interface (metal–oxide interface or the oxide–oxygen interface) will move. Thus, the fixed boundary is unreasonable.

Based on these above, the aim of this article is to investigate the surface oxidation kinetics by adopting perturbation approach without assuming instantaneous reaction at the oxide/oxygen surface. The moving boundary is converted to be the fixed one by Landau transformation. The effects of the reaction, diffusion, and the initial thickness of the oxide layer on the kinetic of the metallic oxidation are discussed.

Analysis

Surface oxidation mechanism of the metals is described, as shown in Figure 1. Suppose that the thermodynamic equilibrium occurs at the surface and the chemical reaction happens at the oxygen/oxidation surface, where the oxidation of metal proceeds to form a fresh oxide. For simplicity, only the first-order chemical reaction is assumed as follows

M + O_{2} \to M O_{2}

(1)

Figure 1.

Scheme for oxidation of the metal.

Suppose that the solution of metal in the metallic oxidation is dilute, that is, $c_{m} < < c_{oxide}$ . The differential mass balance for the metal in the oxide layer satisfies the following

\frac{\partial c_{m}}{\partial t} = D \frac{\partial^{2} c_{m}}{\partial z^{2}}

(2)

where $c_{m}$ is the molar concentration of the metal ion in the oxide, and D denotes the diffusion coefficient of the metal ion.

To each point at the oxide/oxygen surface, the jump mass balances²⁷ for the metal and oxidation require

J_{m} = \frac{r_{m}}{M_{m}} = - k c_{m} z = h (t)

(3)

J_{o x i d e} = \frac{r_{o x y g e n}}{M_{o x y g e n}} = k c_{m} = c_{o x i d e} \frac{d h}{d t}

(4)

where k is the reaction coefficient, $h (t)$ is the thickness of the oxide scale, and $c_{oxide}$ is the molar concentration of the oxide. Equation (4) signifies that the rate of oxidation formed by the reaction at the surface is identical to the rate of oxidation grown over the metal.

In order to solve equation (2), the corresponding initial and boundary conditions are given in the following equations

c_{m} = c_{e q} z = 0

(5)

D \frac{\partial c_{m}}{\partial z} = k c_{m} = c_{oxide} \frac{dh}{dt} z = h (t)

(6)

h = h_{0} t = 0

(7)

where $h_{0}$ is the thickness of the initial oxide scale.

For simplicity, the following dimensionless variables are introduced

C = \frac{c_{m} - c_{eq}}{c_{eq}}, T = \frac{t k^{2}}{D}, Z = \frac{zk}{D}, H = \frac{hk}{D}, H_{0} = \frac{h_{0} k}{D}

(8)

Then, equations (2)–(7) can be rewritten as follows

\frac{\partial C}{\partial T} = \frac{\partial^{2} C}{\partial Z^{2}}

(9)

C = 0 Z = 0

(10)

\frac{\partial C}{\partial Z} = C + 1 Z = H (T)

(11)

\frac{dH}{dT} = \frac{C_{eq}}{C_{oxide}} \frac{\partial C}{\partial Z} Z = H (T)

(12)

Equations (11) and (12) are related to the moving boundary, and the difficulty in solving this problem is that the boundary conditions involving H are an unknown function of time, which is named as the nonlinear boundary. The nonlinearity can be removed by introducing an appropriate transformation proposed by Landau.²⁸ The moving boundary can be transformed into a fixed one through the following Landau transformation

u = \frac{Z}{H}

(13)

which was originally developed to describe the phase change and was mainly applied to remove the nonlinearities in the boundary condition. Using equation (13) and the following chain rule

\frac{\partial C}{\partial Z} = \frac{1}{H} \frac{\partial C}{\partial u}

\frac{\partial^{2} C}{\partial Z^{2}} = \frac{1}{H^{2}} \frac{\partial^{2} C}{\partial u^{2}}

{\frac{\partial C}{\partial T} |}_{Z} = {(\frac{\partial C}{\partial u} \frac{\partial u}{\partial T} + \frac{\partial C}{\partial T}) |}_{u} = (\frac{\partial C}{\partial H} - \frac{u}{H} \frac{\partial C}{\partial u}) \frac{dH}{dT}

\frac{dH}{dT} = {\frac{C_{eq}}{C_{oxide}} \frac{\partial C}{\partial Z} |}_{Z = H (T)} = {\frac{C_{eq}}{C_{oxide}} \frac{1}{H} \frac{\partial C}{\partial u} |}_{u = 1}

Equations (9)–(12) can be transformed into

{ϕ \frac{\partial C}{\partial u} |}_{u = 1} (H \frac{\partial C}{\partial H} - u \frac{\partial C}{\partial u}) = \frac{\partial^{2} C}{\partial u^{2}}

(14)

C = 0 u = 0

(15)

\frac{\partial C}{\partial u} = H (C + 1) u = 1

(16)

\frac{ϕ}{H} \frac{dT}{dH} = {(\frac{\partial C}{\partial u})}^{- 1} u = 1

(17)

H = H_{0} T = 0

(18)

It is noted that equations (16) and (17) become the fixed boundary conditions after Laudau transformation.

In equations (14) and (17), the parameter $ϕ$ is defined as follows

ϕ \equiv \frac{C_{eq}}{C_{oxide}}

(19)

which can stand for the volume variation as described by Pilling and Bedworth²⁶ and Oh.²⁹

Obviously, the equilibrium concentration of the metal ion in the oxidation is much smaller than that of the metallic oxidation in most practical cases, that is, $C_{e q} < < C_{o x i d e}$ . Therefore, the parameter $ϕ$ is much less than 1, that is, $ϕ < < 1$ , which suggests that in a regular perturbation analysis, $ϕ$ can be regarded as the perturbation parameter to solve equation (14).

To facilitate the perturbation solution, the variable of equation (14) is expressed as a power series of the parameter $ϕ$

C (u, H) = C_{0} + ϕ C_{1} + ϕ^{2} C_{2} + \dots

(20a)

Substituting equation (20a) into equation (14) yields

\begin{matrix} {ϕ (\frac{\partial C_{0}}{\partial u} + ϕ \frac{\partial C_{1}}{\partial u} + ϕ^{2} \frac{\partial C_{2}}{\partial u} + \dots) |}_{u = 1} \\ [H (\frac{\partial C}{\partial H} + ϕ \frac{\partial C_{1}}{\partial H} + ϕ^{2} \frac{\partial C_{2}}{\partial H} + \dots) \\ - u (\frac{\partial C}{\partial u} + ϕ \frac{\partial C_{1}}{\partial u} + ϕ^{2} \frac{\partial C_{2}}{\partial u} + \dots)] \\ = \frac{\partial^{2} C_{0}}{\partial u^{2}} + ϕ \frac{\partial^{2} C_{1}}{\partial u^{2}} + ϕ^{2} \frac{\partial^{2} C_{2}}{\partial u^{2}} + \dots \end{matrix}

(20b)

Zeroth-order perturbation

From equation (20b) and the corresponding boundary condition, one can obtain the equation terms of order $ϕ^{0}$

\frac{\partial^{2} C_{0}}{\partial u^{2}} = 0

(21)

C_{0} = 0 u = 0

(22)

\frac{\partial C_{0}}{\partial u} = H (C_{0} + 1) u = 1

(23)

Solving equation (21) consistent with equations (22) and (23) yields

C_{0} = \frac{Hu}{1 - H}

(24)

Substituting equation (23) into equation (17) yields

\frac{ϕ}{H} \frac{dT}{dH} = \frac{1}{2} {(\frac{H}{H - 1})}^{- 1} = \frac{(H - 1)}{2 H}

(25)

Integrating equation (25), one obtains

2 ϕ T = H^{2} - 2 H - H_{0}^{2} + 2 H_{0} \equiv Γ_{0}

(26)

where $H_{0}$ is the dimensionless thickness of the initial oxide layer.

Using equation (8), equation (26) can be changed into the dimensional form as follows

h^{2} + Ah = B (t + τ_{0})

(27)

where

A = - \frac{2 D}{k}, B = 2 ϕ D, τ_{0} = \frac{h_{0}^{2} + A h_{0}}{B}

(28)

Note that the zero-order solution $C_{0} (u, H)$ represents the quasi-steady-state solution which can be obtained from equations (9)–(11) by discarding the time derivation of $C_{0}$ . If the chemical reaction is assumed to be instantaneous, equation (28) reduces to the Clark³⁰ model. If the process of the oxidation is in charge of diffusion, it complies with linear law in a shorter oxidation time, and parabolic law is dominant in a long oxidation time.

First-order perturbation

In the same way, the first-order perturbation solution can be also investigated. Substituting equation (20) into equation (14) and equating terms of order $ϕ^{1}$ , one obtains

{\frac{\partial C_{0}}{\partial u} |}_{u = 1} (H \frac{\partial C_{0}}{\partial H} - u \frac{\partial C_{0}}{\partial u}) = \frac{\partial^{2} C_{1}}{\partial u^{2}}

(29)

C_{1} = 0 u = 0

(30)

\frac{\partial C_{1}}{\partial u} = H C_{1} u = 1

(31)

The solution of equation (29) consistent with equations (30) and (31) is as follows

C_{1} = \frac{H^{3} u}{6 {(1 - H)}^{3}} (\frac{H - 3}{1 - H} + u^{2})

(32)

Again using equation (17), the kinetics of oxidation can be derived as follows

ϕ T = \frac{HdH}{\frac{H}{1 - H} + ϕ [\frac{H^{3}}{2 {(1 - H)}^{3} + \frac{H^{3} (H - 3)}{6 {(1 - H)}^{4}} + \dots}]}

(33)

Integrating equation (33) yields

2 ϕ T = Γ_{0} + ϕ Γ_{1} + \dots

(34)

where

\begin{array}{l} Γ_{1} = H^{2} - H_{0}^{2} + 4 (H - H_{0}) \\ + 6 \ln | \frac{1 - H_{0}}{1 - H} | + 2 (\frac{1}{1 - H_{0}} - \frac{1}{1 - H}) \end{array}

Equation (34) describes the thickness of the oxide layer as a function of time corrected up to the first-order perturbation and can be easily extended to the higher orders. However, due to the limitation of space, we only restrict our analysis up to the first order. The term $ϕ Γ_{1}$ represents the first-order correction to the zero-order solution.

When the initial oxidation thickness is zero, that is, $H_{0} = 0$ , oxidation kinetic equation (34) is rewritten as follows

2 ϕ T = H^{2} - 2 H + ϕ [H^{2} + 4 H - 6 \ln (1 - H) + \frac{2 H}{1 - H}] + …

(35)

Using of equation (8), equation (34) can be changed into the following dimensional form

\begin{matrix} Bt = h^{2} - h_{0}^{2} + A (h - h_{0}) \\ + ϕ [h^{2} - h_{0}^{2} - 2 A (h - h_{0}) + \frac{3}{2} A^{2} \ln \frac{A + 2 h_{0}}{A + 2 h} \\ + A^{3} (\frac{1}{A + 2 h_{0}} - \frac{1}{A + 2 h})] + \dots \end{matrix}

(36)

Seen from equation (36), the oxidation kinetics is obviously related to the chemical reaction coefficient, diffusion coefficient, and the thickness of the initial oxide scale.

In the following, two special cases of equation (36) will be discussed in the dimensional form:

1. If the chemical reaction is instantaneous ( $k \to \infty$ or $A \to 0$ ), equation (36) reduces to

Bt + (1 + ϕ) h_{0}^{2} = (1 + ϕ) h^{2} + \dots

(37a)

that is

h^{2} = \frac{B (t + τ_{0})}{1 + ϕ}

(37b)

where

τ_{0} = \frac{(1 + ϕ) h_{0}^{2}}{B}

(38)

2. If the chemical reaction is instantaneous ( $k \to \infty$ or $A \to 0$ ) and there is no initial oxide layer $(h_{0} = 0)$ , equation (36) reduces to

h^{2} = \frac{Bt}{1 + ϕ} = \frac{2 ϕ D}{1 + ϕ} t

(39)

It is concluded from the two special cases that for instantaneous reaction, the oxide kinetics obeys the parabolic law, no matter whether there exists the initial oxide layer or not, which is consistent with the classical assumption. Moreover, the oxidation kinetic coefficient $k_{p}$ is analytically expressed as $2 ϕ D / (1 + ϕ)$ , and this will provide an experimental method to determine its value.

Discussion

k dependence on oxide thickness

In order to discuss how k influences on the oxide thickness, parameter values are taken as the same order in Oh²⁹ and Peng et al.:³¹ $D = 1.46 \times 10^{- 14} m^{2} / s$ , $h_{0} = 5 \times 10^{- 8} m$ , and $ϕ = 0.3$ . The thickness of the oxide scale for different chemical reaction coefficient k is plotted in Figure 2. It is seen that for the smaller k, the oxidation is controlled by the rate of reaction, and the oxide thickness almost increases linearly. As k increases, the growth rate of the oxide is transformed from linear to parabolic. For the larger k, the instantaneous reaction and the oxidation are controlled by diffusion, and the oxide growth can be represented by a fully parabolic function. As expected in equations (37) and (39), the parabolic rate is reasonable under the assumption of diffusion controlled.

Figure 2.

The variation of oxide thickness for different reaction coefficient k. In this calculation, we have taken $D = 1.46 \times 10^{- 14} m^{2} / s$ , $h_{0} = 5 \times 10^{- 8} m$ , and $ϕ = 0.3$ .

$h_{0}$ dependence on oxide thickness

Let $k = 1.9 \times 10^{- 8} s^{- 1}$ , and the other parameters are assigned the same values as those in section “k dependence on oxide thickness.”Figure 3 describes the effect of the initial oxide thickness $h_{0}$ on the oxidation kinetics and shows that the smaller $h_{0}$ leads to the larger growth rate of oxidation, while the larger $h_{0}$ brings out the smaller rate of oxidation. This means that the growth rate of oxidation becomes slow with the increasing the thickness of the initial oxide scale. It can be seen from Figure (3b) that at the same oxidation time, the velocity of the reactive and the variation of the oxidation thickness become faster when the initial thickness of the scale is thinner.

Figure 3.

The variation of oxide thickness for different $h_{0}$ : (a) h includes $h_{0}$ and the newly generated oxide thickness and (b) $Δ h$ only contains the newly generation oxide thickness. In this calculation, we have taken with parameters $D = 1.46 \times 10^{- 14} m^{2} / s$ , $h_{0} = 5 \times 10^{- 8} m$ , $ϕ = 0.3$ , and $k = 1.9 \times 10^{- 8} s^{- 1}$ .

D dependence on oxide thickness

Figure 4 depicts how the oxide thickness is affected by the diffusion coefficient D. The same parameters are used as in section “k dependence on oxide thickness” except for $k = 1.9 \times 10^{- 8} s^{- 1}$ . It is seen from Figure 4 that the growth rate of the oxide decreases with the increase in diffusion coefficient D. On the other hand, larger diffusion coefficient is associated with smaller the thickness of the oxide layer, that is, lower oxidation. Because in large diffusion case, metal atoms diffuse too fast to react with oxygen at the metal/oxide interface, causing a decreasing rate of oxidation. This means that diffusion has a significant influence on the thickness of oxide.

Figure 4.

The variation of oxide thickness for different D. In this calculation, we have taken $h_{0} = 5 \times 10^{- 8} m$ , $ϕ = 0.3$ , and $k = 1.9 \times 10^{- 8} s^{- 1}$ .

Conclusion

In this article, the moving boundary taking into account volumetric variation during the metal surface oxidation is transformed into the fixed one by Laudau transformation. Then, perturbation approach is employed to analyze the metal surface oxide, and the ratio $ϕ$ is chosen as perturbation parameter. Perturbation results show that the solution of the zeroth-order stands for the quasi-steady state and that of the first-order is instantaneous. If the chemical reaction is instantaneous, our models are reduced to the classical model. Moreover, the kinetic coefficient $k_{p}$ is analytically presented. Finally, the oxide kinetics is discussed for different chemical reaction coefficients, different diffusion coefficients, and different initial thicknesses of the oxide scale. The proposed perturbation scheme can be easily extended to the higher order.

Footnotes

Academic Editor: Xiaotun Qiu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is supported by NSFC (grant no. 11402054), Natural Science Basic Research Plan in Shaanxi Province of China (nos 2015JM1011 and 2016JQ1032), Scientific Research Program Funded by Shaanxi Provincial Education Commission (no. 16JK1504), 2016 Open Projects of Key Laboratory for Strength and Vibration of Mechanical Structures (no. SV2016-KF-18) and Project Funded by China Postdoctoral Science Foundation (no. 2015M570552).

References

Golmon

Maute

Dunn

ML.

Numerical modeling of electrochemical-mechanical interactions in lithium polymer batteries. Comput Struct 2009; 87: 1567–1579.

Shenoy

Unnam

Clark

RK.

Oxidation and embrittlement of Ti-6Al-2Sn-4Zr-2Mo alloy. Oxid Met 1986; 26: 105–124.

Evans

HE.

Stress effects in high temperature oxidation of metals. Int Mater Rev 1995; 40: 1–40.

Imbrie

Lagoudas

DC.

The morphological evolution of TiO₂ scale formed on various 1D and 2D geometries of Titanium. Oxid Met 2001; 55: 359–399.

Zhao

Pharr

Vlassak

. Inelastic hosts as electrodes for high-capacity lithium-ion batteries. J Appl Phys 2011; 109: 016110.

Saillard

Cherkaoui

Capolungo

. Stress influence on high temperature oxide scale growth: modeling and investigation on a thermal barrier coating system. Philos Mag 2010; 90: 2651–2676.

Wagner

Atom movements. Cleveland, OH: American Society for Metals, 1951, pp.153–173.

Kim

Moorhead

AJ.

Oxidation behavior and flexural strength of aluminum nitride exposed to air at elevated temperatures. J Am Ceram Soc 1994; 77: 1037–1041.

Bellosi

Landi

Tampieri

Oxidation behavior of aluminum nitride. J Mater Res 1993; 8:565–572.

10.

Osborne

Norton

MG.

Oxidation of aluminum nitride. J Mater Sci 1999; 33: 3859.

11.

Chou

KC.

A kinetics model for oxidation of Si-Al-O-N materials. J Am Ceram Soc 2006; 89: 1568.

12.

Chou

Lin

. Kinetics of absorption and desorption of hydrogen in alloy powder. Int J Hydrogen Energ 2005; 30: 301–307.

13.

ES.

A diffusional analysis for the oxidation on a plane metal–oxide interface. Chem Eng J 2006; 117: 143–154.

14.

Suo

Kubair

Evans

. Stresses induced in alloys by selective oxidation. Acta Mater 2003; 51: 959–974.

15.

Entchev

Lagoudas

Slattery

JC.

Effects of non-planar geometries and volumetric expansion in the modeling of oxidation in titanium. Int J Eng Sci 2001; 39: 695–714.

16.

Kofstad

Anderson

Krudtaa

OJ.

Oxidation of titanium in the temperature range 800–1200°C. J Less-Common Met 1961; 3: 89–97.

17.

Smeltzer

WW.

The influence of short-circuit grain boundary diffusion on the growth of oxide layers on metals. Mater Sci Forum 1988; 29: 151–172.

18.

Kofstad

High-temperature corrosion. London: Elsevier, 1988.

19.

Kofstad

Nonstoichiometry, diffusion and electrical conductivity in binary metal oxides. New York: John Wiley & Sons, 1972.

20.

Shen

. Non-equilibrium thermodynamics and variational principles for fully coupled thermal-mechanical-chemical processes. Acta Mech 2013; 224: 2895–2910.

21.

Zhang

. Analytical modeling on stress assisted oxidation and its effect on creep response of metals. Oxid Met 2014; 82: 311–330.

22.

Zhou

Cherkaoui

Stress-oxidation interaction in selective oxidation of Cr-Fe alloys. Mech Mater 2010; 42: 63–71.

23.

Dong

Fang

Feng

. Diffusion and stress coupling effect during oxidation at high temperature. J Am Ceram Soc 2013; 96: 44–46.

24.

Maharjan

Zhang

Wang

. Residual stresses within oxide layers due to lateral growth strain and creep strain: analytical modeling. J Appl Phys 2011; 110: 063511.

25.

Ruan

Pei

Fang

DN.

Residual stress analysis in the oxide scale/metal substrate system due to oxidation growth strain and creep deformation. Acta Mech 2012; 223: 2597–2607.

26.

Pilling

Bedworth

RE.

The oxidation of metals at high temperature. J I Met 1923; 29: 529–591.

27.

Slattery

JC.

Advanced transport phenomena. 1st ed. Cambridge: Cambridge University Press, 1999.

28.

Landau

HG.

Heat conduction in a melting solid. Q Appl Math 1950; 8: 81–94.

29.

ES.

A perturbation analysis for the metal oxidation of cylindrical geometries. J Chem Eng Jpn 2006; 39: 57–67.

30.

Clark

DR.

The lateral growth strain accompanying the formation of a thermally grown oxide. Acta Mater 2003; 51: 1393–1407.

31.

Peng

Wang

Slattery

JC.

A new theory for silicon oxidation. J Vac Sci Technol B 1996; 14: 3316–3320.

Perturbation approach to metal surface oxidation considering volume variation

Abstract

Keywords

Introduction

Analysis

Zeroth-order perturbation

First-order perturbation

Discussion

k dependence on oxide thickness

h 0 dependence on oxide thickness

D dependence on oxide thickness

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References

$h_{0}$ dependence on oxide thickness