Onsager Diffusion 9.2

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Theory & Documentation

Onsager Multi-Component Diffusion Solver • v9.2

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 the gradients of all concentrations, split into kinetic cross-coupling and thermodynamic driving forces:

$$ J_i = - \sum_{j} \sum_{k} D_{ik}^{kin} \Phi_{kj} \nabla C_j $$
  • $D_{ik}^{kin}$: Onsager-Manning kinetic coefficients (Vacancy Wind / Site sharing).
  • $\Phi_{kj}$: Thermodynamic factor matrix derived from the $\Omega$-matrix.

2.2 Kinetics: The Q-Matrix & Vacancies

The local tracer diffusivity depends on the concentration-weighted activation barrier ($Q$) and the dynamic vacancy fraction ($V_{factor}$):

$$ D_i^*(x) = D_{0,i}\, \exp\left(-\frac{\sum_j C_j(x)\, Q_{ij}}{k_B T}\right) \times V_{factor}(x) $$

Note: $V_{factor}$ is calculated dynamically. Aliovalent donors (Valence > 2) increase vacancies, accelerating diffusion. Acceptors (Valence < 2) suppress free vacancies down to the intrinsic Schottky limit.

2.3 Thermodynamics: The Ω-Matrix

The $\Omega$-matrix defines enthalpic mixing interactions using a regular-solution model ($ H_{mix} = \sum \Omega_{ij} X_i X_j $). Positive $\Omega$ implies demixing/clustering. This is converted into the thermodynamic matrix $\Phi$:

$$ \Phi_{ij} = \delta_{ij} - \frac{2 \Omega_{ij} C_i C_j}{k_B T} $$

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.

IonD₀ (m²/s)Q (eV)Notes / Ref Context
Mg²⁺ 1.0e‑10 2.30 Host Matrix. Wuensch et al. (1973).
Fe²⁺8.0e-101.76 Fast Blank & Pask (1969).
Zn²⁺8.0e-101.85 Fast Wuensch (1983).
Mn²⁺1.5e-102.10 Intermediate speed.
Co²⁺4.0e-102.06 Chen & Peterson (1980).
Cu²⁺3.0e-102.15 Jahn-Teller distortion.
Ni²⁺1.0e-102.28 Ideal Hardin et al. (1973).
Ca²⁺0.5e-102.70 Slow Size Mismatch ($1.00\mathring{A}$).
V³⁺6.0e-102.35 Aliovalent defect drag.
Sc³⁺4.0e-102.40 Early 3d series.
Ga³⁺4.5e-102.30 Post-transition.
Fe³⁺5.0e-102.45 Gourdin et al. (1979).
Cr³⁺3.5e-102.50 Weber (1972).
Co³⁺2.0e-102.55 High electrostatic interaction.
Ti⁴⁺2.0e-102.80 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⁺0.01Ideal Solution
Mg - Ni0.02Ideal Solution
Mg - Co²⁺0.02Ideal Solution
Fe²⁺ - Co²⁺0.02Near Ideal
Mg - Fe²⁺0.03Near Ideal
Mg - Mn0.04Low Mismatch
Mg - Cu0.15Jahn-Teller Strain
Ni - Zn0.20Structural Mismatch
Mg - Zn0.25Tetrahedral Preference
Mg - Sc0.45Charge Mismatch
Mg - Ga0.45Charge Mismatch
Mg - Fe³⁺0.50Defect Clustering
Mg - V0.50Defect Clustering
Mg - Co³⁺0.50Defect Clustering
Mg - Cr0.55Defect Clustering
Mg - Ca0.65Large Miscibility Gap
Mg - Ti0.70Strong 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.