Final Concentration Profile (2D) t = Final
Concentration Evolution (3D)
Final Figure of Merit zT (2D) Avg ZT: -
ZT Evolution (3D)
Final Seebeck Coefficient (2D)
Seebeck Evolution (3D)
Final Conductivity (2D)
Conductivity Evolution (3D)
User Guide
ThermoMigrate v3.4 Documentation
How to use "Smart Input"
The Concentration field accepts JavaScript math expressions. You can define complex initial gradients to test Functionally Graded Materials (FGM).
The Variable 'x'
-
x = 0.0Hot Side (Left) -
x = 1.0Cold Side (Right)
The profile is normalized to leg length.
Supported Math
Standard JS operators (+, -, *, /) apply.
Exemple Profile Equations
Copy and paste these directly into the Concentration input field to test different scenarios.
| Profile Type | Equation Example | Why test this? |
|---|---|---|
| Constant (Flat) | 1e19 |
The standard baseline. |
| Linear Grading |
2e19 * (1 - 0.5*x)
Starts at 2e19, drops to 1e19
|
Optimizes $zT$ by matching carrier conc to temperature ($n \propto T^{1.5}$). |
| Exponential | 1e19 * exp(2*x) |
Rapidly increasing dopant towards the cold side. |
| Step Function |
x < 0.5 ? 2e19 : 1e18
|
Simulates a diffusion couple (High conc joined to Low conc). |
| Parabolic | 1e19 * (1 + 4*x*(1-x)) |
Peak concentration in the middle of the leg. |
3. Underlying Physics (v2 Model)
A. The Soret Effect
Dopants migrate due to the temperature gradient. The direction is determined by the Heat of Transport ($Q^*$).
- If Q* > 0: Dopants move to the COLD side.
- If Q* < 0: Dopants move to the HOT side (Thermophilic).
B. High-Temperature Accuracy (Bipolar)
To prevent unrealistic $zT$ predictions at high temperatures (where standard models fail), this simulation uses a Bipolar Correction calibrated to p-type PbSe data.
- Effective Band Gap: We use $E_g \approx 0.28$ eV to calculate minority carrier activation.
- Seebeck Suppression: As $T$ rises, the Seebeck coefficient is reduced by a factor of $\frac{1}{1 + 5\xi}$, where $\xi$ is the bipolar factor. This simulates the "rollover" seen in real experiments.
- Thermal Conductivity: An extra term $\kappa_{bi}$ is added at high $T$, further lowering the $zT$ to realistic limits ($zT_{max} \approx 1.2$).
C. Numerical Solver
The tool solves the 1D Continuity Equation using an explicit Finite Difference scheme. Time steps ($\Delta t$) are automatically adjusted to satisfy the CFL stability condition ($\Delta t < 0.4 \Delta x^2 / D_{max}$), ensuring the simulation remains stable even with fast-diffusing dopants (like Ag or Cu).