Numerical Implementation and Application of an Extended Model for Diffusion and Precipitation of Chemical Elements in Metallic Matrices

Abstract

A model for simultaneous and coupled diffusion and precipitation of chemical elements in metallic matrices is proposed, together with a scheme for its numerical solution and some applications to problems of internal oxidation. This model extends over a previous one of the authors, which itself stood as a generalization of many earlier ones. It includes the possibility that precipitates may be present in large fractions; if so, they may affect diffusion in the matrix phase by acting as “barriers”, and/or by distorting this matrix. The possible re-dissolution of some or all precipitates may also be prohibited for kinetic reasons. A simple but robust numerical scheme based on 1D finite differences in space coupled with explicit integration in time is proposed to solve the equations of the problem. This scheme notably contains an efficient Newton–Raphson algorithm used to solve the laws of mass action. Numerical simulations of internal oxidation processes close to those encountered in actual industrial practice are finally presented. It is first shown that in simple cases, the simulations reproduce Wagner’s classical analytical solution and related ones. More complex cases not amenable to analytic solutions are then envisaged.

Appendices

Appendix 1: Diffusion and Precipitation Data

The atomic masses (in g mol⁻¹) of O, Al, Mn and Si and their diffusion coefficients (in μm² s⁻¹) in pure iron are provided by the following formulae, where the absolute temperature T is in K:

$$ \left\{ \begin{array}{l} M_{\rm O} = 15.999 \\ M_{\rm Al} = 26.982 \\ M_{\rm Mn} = 54.938 \\ M_{\rm Si} = 28.086 \end{array} \right.; \quad \left\{ \begin{array}{l}D_{\rm O}^0 = 2.44\; 10^{5} \exp(-11078/T) \\ D_{\rm Al}^0 = 5.15 \;10^{8} \exp(-29566/T) \\ D_{\rm Mn}^0 = 7.56\; 10^{7} \exp(-26983/T) \\ D_{\rm Si}^0 = 7.35\; 10^{7} \exp(-26439/T). \end{array} \right. $$

(16)

Expressions (16)_5,6,7,8 for the D ⁰ _i here are taken from the work of Oikawa [22].

The molecular masses (in g mol⁻¹) of Al₂O₃, MnO, SiO₂ and Mn₂SiO₄, their solubility products (in ppmⁿ where n is the total number of atoms in the molecule), and their specific volumes (in cm³ g⁻¹) together with that of pure iron, are given by

$$ \left\{ \begin{array}{l} M_{{\rm Al}_2{\rm O}_3} = 101.961 \\ M_{\rm MnO} = 70.937 \\ M_{{\rm SiO}_2} = 60.084 \\ M_{{\rm Mn}_2{\rm SiO}_4} = 201.958 \end{array}\right.; \left\{ \begin{array}{l} {\rm log}_{10} K_{{\rm Al}_2{\rm O}_3} = 28.55-49600/T \\ {\rm log}_{10} K_{\rm MnO} = 10.95-10830/T \\ {\rm log}_{10} K_{{\rm SiO}_2} = 16.00-22010/T \\ {\rm log}_{10} K_{{\rm Mn}_2{\rm SiO}_4} = 38.56-46870/T \\ \end{array}\right.; \left\{ \begin{array}{l} v_{\rm Fe} = 0.1272 \\ v_{{\rm Al}_2{\rm O}_3} = 0.2518 \\ v_{\rm MnO} = 0.1862 \\ v_{{\rm SiO}_2} = 0.3776 \\ v_{{\rm Mn}_2{\rm SiO}_4} = 0.2433. \end{array}\right. $$

(17)

Formulae (17)_5,6,7,8 for the K _α are taken from the Thermodata [27] electronic bank, except that for K _{Mn 2}SiO₄ which was adjusted using experimental data provided by Eriksson et al. [6].

Appendix 2: Dependence of the Solution Upon the Sole Variable $x/ \sqrt{t}$

In order to prove that the solution of a given 1D diffusion problem depends on the sole variable

$$ u \equiv \frac{x}{\sqrt{t}}, $$

(18)

one must rewrite the equations of this problem with this assumption, and check that no contradiction then arises; this means that all quantities appearing in the new equations must depend on u alone.

Initial Model

We first study the case of the “initial” model which includes neither the reduction of diffusion coefficients in Eq. 3 nor the dilation of coordinates in Eq. 5, and considers re-dissolution of precipitates as possible. We assume all quantities F ^T _i and F ^D _i (i = 1, 2,…n _e ) and P _α (α = 1, 2,…n _p ) to be functions of the sole variable u. This does not raise any problem in the balance Eq. 1, nor in the laws of mass action (6) (with the “ALLOW” option). Also, Eqs. 2 take the form

$$ \frac{u}{2} \frac{dF_i^T}{du}(u) + D_i \frac{d^2F_i^D} {du^2}(u) = 0 \quad (i=1,2,\ldots n_e), $$

(19)

and there is again no problem since all quantities here depend only on u. The boundary conditions discussed in the section on “Treatment of Boundary Conditions” read

$$ \left\{ \begin{array}{lll} F^D_1(u=0) = (F^D_1)^{\rm surf}; &\frac{\partial F_i^D} {\partial u}(u=0) = 0 & \quad (i=2,3,\ldots n_e) \\ F_1^D(u=+\infty) = 0; & F_i^D(u=+\infty) = (F_i^D)^{\rm core} & \quad (i=2,3,{\ldots}n_e), \end{array}\right. $$

(20)

and the initial conditions read

$$ \left\{ \begin{array}{ll} F_1^T(u=+\infty) = F_1^D(u=+\infty) = 0 \\ F_i^T(u=+\infty) = F_i^D(u=+\infty) = (F_i^D)^{\rm core} &(i=2,3,\ldots n_e) \\ P_{\alpha}(u=+\infty) = 0 & (\alpha=1,2,\ldots n_p). \end{array}\right. $$

(21)

These conditions are non-contradictory (note that Eq. 21 _1,2 duplicate Eq. 20 ₂), and again involve the sole variable u. Thus all equations of the problem are compatible with a solution depending only on the variable u.

Model with Reduction of Diffusion Coefficients

The introduction of the possible reduction of diffusion coefficients by precipitation (Eq. 3) modifies only Eqs. 2. Let ϕ(u) denote the ratio in the right-hand side of Eq. 3. (This is a function of u alone since the P _α (α = 1, 2,…n _p ) are assumed to depend on this sole variable). The diffusion equations then take the form

$$ \frac{u}{2} \frac{dF_i^T}{du}(u) + D_i^0 \frac{d}{du} \left[ \phi(u) \frac{dF_i^D}{du}(u) \right] = 0 \quad (i=1,2,\ldots n_e), $$

(22)

which is again compatible with the assumption that the solution depends only on u.

Model with Dilation of Coordinates

When the possible dilation of coordinates by precipitation (Eq. 5) is accounted for, one must distinguish between the initial coordinate, x ⁰, and the present one, x. We then define the variable u as

$$ u \equiv \frac{x^0}{\sqrt{t}}. $$

(23)

Again, the dilation of coordinates modifies only Eqs. 2. By Eq. 5,

$$ \frac{\partial x^0}{\partial x} = \phi(u) \quad \Rightarrow \quad \frac{\partial}{\partial x}= \phi(u) \frac{\partial}{\partial x^0} $$

where ϕ(u) is the same function as before. Therefore the diffusion equations take the form

$$ \dot{F_i^T} = D_i \phi(u) \frac{\partial}{\partial x^0} \left[ \phi(u) \frac{\partial F_i^D}{\partial x^0} \right] \quad (i=1,2,\ldots n_e) $$

(where the dot denotes the material time-derivative, that is the partial differentiation with respect to t at constant x ⁰), or equivalently

$$ \frac{u}{2} \frac{dF_i^T}{du}(u) + D_i \phi(u) \frac{d} {du} \left[ \phi(u) \frac{dF_i^D}{du}(u) \right] = 0 \quad (i=1,2,\ldots n_e). $$

(24)

This equation is again compatible with the assumption that the solution depends only on u.

Model with Prohibition of Re-Dissolution of Precipitates

We finally consider the model where possible re-dissolution of precipitates is prohibited. This prohibition only modifies the laws of mass action (6), where the option “FORBID” must now be considered. Then the conditions P _α = 0 in Eq. 6 ₁ and P _α > 0 in Eq. 6 ₃ must be replaced by $\dot{P}_{\alpha}=0,$ $\dot{P}_{\alpha}>0.$ But if P _α depends only on u, these conditions may be written in the form $\frac{dP_{\alpha}}{du}(u)=0,$ $\frac{dP_{\alpha}} {du}(u)<0.$ Hence the laws of mass action may again be written in a form involving the sole variable u, which again shows that the solution depends on this sole variable.