Theory & Documentation
Onsager Multi-Component Diffusion Solver • v9.0
1. Overview
This tool simulates multi-component cation diffusion in oxide crystals using the Onsager transport formalism. Unlike simple Fickian diffusion where species diffuse independently, real crystals exhibit correlated motion. Every jump requires a vacancy, and vacancies are shared by all species, producing cross-coupling between fluxes.
2. Physical Models
2.1 Onsager Flux Equation
For each species i, the flux is governed by gradients of all concentrations:
- Diagonal terms: diffusion of species i down its own gradient.
- Off‑diagonal terms: cross‑coupling due to vacancy sharing (Vacancy Wind).
2.2 Kinetics: The Q‑Matrix
The effective tracer diffusivity used in the solver is:
2.3 Thermodynamics: The Ω‑Matrix
The Ω‑matrix defines enthalpic mixing interactions using a regular‑solution model ($ H_{mix} = \sum \Omega_{ij} X_i X_j $). Positive $\Omega$ implies demixing/clustering.
3. Reference Values (MgO Host Matrix)
The defaults below are the exact values used in the simulation code. $D_0$ values assume the extrinsic vacancy regime ($D \approx 10^{-10} m^2/s$).
3.1 Complete Cation Database (D₀ & Q)
Full list of cations available in the "Dopants" dropdown.
| Ion | D₀ (m²/s) | Q (eV) | Notes / Ref Context |
|---|---|---|---|
| Mg²⁺ | 1.0e‑10 | 2.30 | Host Matrix. Wuensch et al. (1973). |
| Fe²⁺ | 8.0e-10 | 1.76 | Fast Blank & Pask (1969). |
| Zn²⁺ | 8.0e-10 | 1.85 | Fast Wuensch (1983). |
| Mn²⁺ | 1.5e-10 | 2.10 | Intermediate speed. |
| Co²⁺ | 4.0e-10 | 2.06 | Chen & Peterson (1980). |
| Cu²⁺ | 3.0e-10 | 2.15 | Jahn-Teller distortion. |
| Ni²⁺ | 1.0e-10 | 2.28 | Ideal Hardin et al. (1973). |
| Ca²⁺ | 0.5e-10 | 2.70 | Slow Size Mismatch ($1.00\mathring{A}$). |
| V³⁺ | 6.0e-10 | 2.35 | Aliovalent defect drag. |
| Sc³⁺ | 4.0e-10 | 2.40 | Early 3d series. |
| Ga³⁺ | 4.5e-10 | 2.30 | Post-transition. |
| Fe³⁺ | 5.0e-10 | 2.45 | Gourdin et al. (1979). |
| Cr³⁺ | 3.5e-10 | 2.50 | Weber (1972). |
| Co³⁺ | 2.0e-10 | 2.55 | High electrostatic interaction. |
| Ti⁴⁺ | 2.0e-10 | 2.80 | Very Slow High charge drag. |
3.2 Interaction Energies ($\Omega$)
Explicit pair values defined in the code. Pairs not listed here use the Default (0.05 eV).
| Defined Pair | $\Omega$ (eV) | Physical Origin |
|---|---|---|
| Ni - Co²⁺ | 0.01 | Ideal Solution |
| Mg - Ni | 0.02 | Ideal Solution |
| Mg - Co²⁺ | 0.02 | Ideal Solution |
| Fe²⁺ - Co²⁺ | 0.02 | Near Ideal |
| Mg - Fe²⁺ | 0.03 | Near Ideal |
| Mg - Mn | 0.04 | Low Mismatch |
| Mg - Cu | 0.15 | Jahn-Teller Strain |
| Ni - Zn | 0.20 | Structural Mismatch |
| Mg - Zn | 0.25 | Tetrahedral Preference |
| Mg - Sc | 0.45 | Charge Mismatch |
| Mg - Ga | 0.45 | Charge Mismatch |
| Mg - Fe³⁺ | 0.50 | Defect Clustering |
| Mg - V | 0.50 | Defect Clustering |
| Mg - Co³⁺ | 0.50 | Defect Clustering |
| Mg - Cr | 0.55 | Defect Clustering |
| Mg - Ca | 0.65 | Large Miscibility Gap |
| Mg - Ti | 0.70 | Strong Repulsion ($4+$) |
4. Bibliography
- Blank, S. L., & Pask, J. A. (1969). "Diffusion of Iron and Nickel in Magnesium Oxide." J. Am. Ceram. Soc. 52.
- Chen, W. K., & Peterson, N. L. (1980). "Grain-boundary diffusion of 60Co and 51Cr in NiO." J. Am. Ceram. Soc. 63.
- Doman, R. C., et al. (1973). "Phase Equilibria in the System CaO-MgO." J. Mater. Sci. 8.
- Gourdin, W. H., & Kingery, W. D. (1979). "The defect structure of MgO containing trivalent cation solutes." J. Mater. Sci. 14.
- Hardin, B. D., et al. (1973). "Diffusion of Ni2+ in MgO." J. Am. Ceram. Soc. 56.
- Rungis, J., & Mortlock, A. J. (1966). "The diffusion of Calcium in Magnesium Oxide." Phil. Mag. 14.
- Weber, G. W. et al. (1972). "Diffusion of 51Cr in MgO." J. Chem. Phys.
- Wuensch, B. J., et al. (1973). "Diffusion of Mg-28 in MgO up to 1900C." J. Chem. Phys. 58.