Course: Materials Simulation Practical | FAU Erlangen-Nürnberg
Tools: Python · FiPy · NumPy · Matplotlib
📓 Written report | 💻 Simulation code
Numerical implementation of the phase-field method using the finite volume approach in FiPy. The project covers two coupled formulations: (1) the Allen-Cahn equation for non-conserved order parameter evolution, applied to 1D solidification with systematic parameter studies; and (2) the Cahn-Hilliard equation for conserved concentration dynamics, applied to spinodal decomposition in the Ni-Al binary alloy system in both 1D and 2D.
The Allen-Cahn equation governs the time evolution of a non-conserved order parameter φ ∈ [0,1], where φ = 1 represents the solid phase and φ = 0 the liquid. The system is initialized with a sharp step interface at x = 100 on a domain of length 200, using a double-well bulk free energy f(φ) = f₀ φ²(1−φ)².
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| Order parameter profile at steady state | Total free energy decay vs. time |
The diffuse interface profile shows the characteristic hyperbolic tangent shape, with interface width controlled by the competition between the gradient penalty K_φ and the bulk energy barrier f₀. The total free energy F(t) decays monotonically to its equilibrium value, confirming thermodynamic consistency. The rapid initial drop corresponds to sharp-interface relaxation; the slow tail reflects diffuse interface equilibration.
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| Free energy density vs. x at steady state — gradient contribution dominates over bulk at the interface |
In the final state, the free energy density is localized entirely at the interface. The gradient contribution f_grad dominates over the bulk contribution f_bulk, consistent with a well-resolved diffuse interface where the bulk phases are fully relaxed.
The effect of three model parameters on the interface profile and energy landscape was systematically investigated by varying one parameter at a time while holding the others fixed.
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| Effect of f₀ — higher values sharpen the interface and raise total stored energy without affecting interface position |
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| Effect of K_φ — increasing K_φ widens the interface and increases gradient energy cost while leaving bulk energy largely unaffected |
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| Effect of L — controls kinetics only; all curves converge to the same final φ(x) profile, with higher L accelerating the approach to equilibrium |
The Cahn-Hilliard equation governs conserved concentration dynamics via:
∂c/∂t = ∇ · (M ∇μ), μ = df/dc − K_c ∇²c
Physical parameters for the Ni-Al system:
| Parameter | Value |
|---|---|
| c_γ (Al-poor equilibrium) | 0.16 |
| c_γ' (Al-rich equilibrium) | 0.23 |
| Gradient coefficient K_c (1D) | 3.9×10⁻⁶ J/m |
| Gradient coefficient K_c (2D) | 2.45×10⁻⁷ J/m |
| Mobility M | 10⁻¹⁷ · Vm² m⁵/(J·s) |
| Bulk energy scale f₀ | 9.989×10⁷ J/m³ |
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| Final 1D concentration profile — sharp segregation between γ (c = 0.16) and γ' (c = 0.23) phases |
The final concentration profile shows sharp segregation between the γ (Al-poor, c = 0.16) and γ' (Al-rich, c = 0.23) phases, with a well-resolved diffuse interface. Computed interfacial energies at steady state:
| Contribution | Value |
|---|---|
| F_bulk | 0.2367 J/m² |
| F_grad | 0.2368 J/m² |
| F_total | 0.4735 J/m² |
The near-equal bulk and gradient contributions confirm the interface is at its natural width set by the K_c / f₀ balance.
Starting from a Gaussian noise initial condition centered at c₀ = 0.195 on a 100×100 nm² domain, the 2D simulation captures the full spinodal decomposition process from a disordered solid solution into phase-separated γ and γ' domains. Over time, diffuse Al-poor and Al-rich domains nucleate and coarsen, driven purely by free energy minimization.
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| 2D spinodal decomposition — Ni-Al system, 100×100 nm², 1000 time steps |
pip install fipy numpy matplotlib scipyOpen phase_field_simulation.ipynb in Jupyter and run cells sequentially. The 2D Cahn-Hilliard simulation runs 1000 steps on a 100×100 nm² mesh and saves the animation as ch_2d.gif. Expect 10–20 minutes on a standard laptop.