Lecture 9: Kinetics of Oxidat of Metals: Part : Wagner Parabolic Mod Today s topics Oxidat of metals: controlled by both the ic diffus (carried by M + and O - ) and the ectronic diffus (carried by e - and h + ). Combinat of the ic diffus and ectronic diffus gives a net result of diffus that apparently shows the diffus of molecular oxygen (O ) from the air/ interface to the /M interface, and the reverse for the diffus of metal atoms. This way maintains the Nernst potential caused by the difference of partial pressure of oxygen at the two interfaces. Understanding of the Wagner s parabolic mod that describes the rate of growth of the layer: implicats for how to inhibit the growth of oxide for the purpose of protecting the metals from corros. Last Lecture: Since the metal oxide is of ic lattice, the diffus across the oxide layer (that controls the growth of the oxide phase) should be carried out by s, M + and/or O -, rather than the neutral atoms of metal or molecules of oxygen, as depicted in the diagram bow. Air () () O - M + M + + O - /O + e O - M + + O - h + e - M M + + e +V, Nernst Potential, created by difference in partial pressure of Oxygen M The total ic current density caused by the diffus of both s is: = + + = ( + ) + E = E = dφ dx As can be seen from the diagram above, there is a gradient in po across the oxide film, where at the air/
() interface P 0.atm (with air as the atmosphere), and at the M/ interface P O () O << atm. This creates () RT PO a so-called Nernst potential across the film given by V = ln{ } () 4F P. This potential (with the air/ side more positive) represents a chemical equilibrium established by the reverse diffus of M + and O - as depicted in the diagram above. O Assuming if only ic current flows through the oxide film, the transport will quickly stop as soon as enough M + s are transported to /air interface, and/or enough O - s are transported to M/ interface to () RT PO reach the equilibrium as defined by the Nernst Potential V = ln{ } () 4F P as mented above. However, that the oxide film also conducts ectrons and/or ectron holes, and flow of these ectrons and holes neutralize the flow of the M + and O - s --- the overall result combining these two pairs of diffus is that O moves to the M/ interface and M moves to the air/ interface, to keep the oxide layer growing and maintain the Nernst Potential (i.e., the thermodynamic driving force for the phase growth) until the oxide layer becomes too thick, i.e., the diffus across the whole layer takes forever (now becoming Kinetics limited). O Today s Lecture: Continue to discuss the diffus flux involving the ectrons and holes, e - and h +. Combining this diffus pair to the diffus pair of M + and O - s leads to the Wagner s parabolic rate law, which describes the kinetics (rate) of the layer growth. dϕ dϕ The ectronic current density is given by = = ( e + h) dx dx Where e = ectron conductivity h = hole conductivity = total ectronic conductivity ϕ = ectrochemical potential of ectrons divided by the ectronic charge Now, there is no externally applied voltage. Therefore, the net current through the film must be zero. That is + = 0 To preserve ectroneutrality. dφ dϕ Thus, + = 0 dx dx The overall process can be schematically described as the following schematics:
Air () () O - M + M + + O - /O + e O - M + + O - h + e - M M + + e +V, Nernst Potential, created by difference in partial pressure of Oxygen M The above figure shows that for every one O - moving, h + move in the opposite direct; and for every one M + moving, e - move in the same direct. The net result is that there is no net current. However, there is the so-called internal current. The above can be represented by an equivalent circuit: R + - V R The internal current density, int = R V + R e l, where V is the Nernst potential R = onic resistance per unit area = x /, where x is the thickness of metal oxide. R = ectronic resistance per unit area = x / Then, V V int = = x x + ( + ) x Also we have, int = () dt Where, = charge (internal) transported in time dt () When /unit area ic charge is passed through the film (and of course an equal amount of ectronic charge also passed through), the molar number of /unit area to be formed can be written as: 3
dn = F where F: Faraday constant, = 96487 coulombs/mole = e N 0 =.6 0-9 6.0 0 3 (see Lecture 9) f W is the molecular weight of, then Wdn = W (with unit of gram/cm ) F f ρ is the density of, then W W dn ρ = ρ F, is the volume of formed (created) per unit area Since volume = unit area (.0) dx (change in thickness) = dx So, Or, W = dx ρ F Fρ = dx (3) W Substituting Eq. (3) into Eq. (), we have int Fρ dx = = (4) dt W dt Comparing Eq. (4) and Eq. (), we have int = Fρ dx V W dt = ( + ) x then, WV xdx = dt Fρ ( + ) ntegrat of the above equat gives: x WV = t Fρ ( + ) (5) Eq. (5) can be re-written in the format of 4
x = Kt (6) Where, W V K =, is the oxidat rate constant, in unit of length /second, e.g., cm /sec. F ρ ( + ) Eq. (6) is known as the Wagner s parabolic rate law, or Wagner s parabolic mod. The rate constant, K, contains both the thermodynamic and the kinetic factors. The thermodynamic factor is embodied in the Nernst potential V, which is rated to the free energy for the react: M + O The kinetic factor is embodied in the term + Note that if either 0 or 0 f >>, then the rate constant is, the kinetics of oxidat will be very slow. K W Fρ V f >>, then K W Fρ V Supposing the latter situat ( >> ) describes the kinetics of oxidat, since = + + Then, the larger of the two will dictate the kinetics. W Supposing + >>, then, the rate constant is K + V M Fρ 4F Since M + = C D M + M +, where N 0 is the Avagadro constant. RTN 0 5
4FW Then: K C + D + V M M RTN ρ 0 The suppress of the kinetics of oxidat of metals and alloys thus would require that the diffus coefficient of M + in the oxide be suppressed to prevent the oxidat of a given metal or alloy. Some following up remarks about oxidat of metals:. Transport properties of oxides, halides, sulfides, carbonates, etc, can be changed by doping with s of a valence different from the host the so-called aliovalent dopant ( alio means different). This is similar to case of semiconductor materials wherein the concentrats of ectrons and holes are altered by introducing aliovalent dopants.. The general theory of oxidat of metals is equally applicable to halides, sulfides, sulfates, carbonates, nitrates, etc.. 3. Most super alloys based on Fe, Ni, and Co contain other ements to improve the oxidat resistance. For example, many of the alloys contain Al or Cr. The partial pressure (required P ) for the oxidat of Al to Al O 3 and Cr to Cr O 3 is much lower than for Fe to FeO (Fe O 3 ), Ni to NiO and Co to CoO. The oxidat resistance of the super-alloys is enhanced by the format of a protective layer of Al O 3 or Cr O 3 on the surface. O 6