ShaneLogic/boundary-conditions-and-non-dimensionalisation

Boundary Conditions and Non-dimensionalisation

Use this skill when defining the solution space for PDE drift-diffusion models or scaling variables for numerical stability in perovskite solar cell simulations.

Apply Boundary Conditions

At x = 0 (ETL interface):

1. Anion vacancies: No flux condition - F_P = 0
2. Potential: Specified as φ = V_ap (applied voltage)
3. Electron concentration: n = n_b (determined by band offsets)
4. Hole current: j_p = R_l (surface recombination rate at left interface)

At x = b (HTL interface):

1. Anion vacancies: No flux condition - F_P = 0
2. Potential: Specified as φ = V_bi (built-in potential)
3. Hole concentration: p = p_b (determined by band offsets)
4. Electron current: j_n = -R_r (surface recombination rate at right interface)

Define Initial Conditions

Set initial profiles for:

• n(x,0) = n_i(x) - electron density
• p(x,0) = p_i(x) - hole density
• P(x,0) = P_i(x) - vacancy density

Perform Non-dimensionalisation (Scaling)

Spatial Scaling

• x* = x / b
• Where b = perovskite layer width

Time Scaling

• t* = t / τ_ion
• Where τ_ion = L_d * b / D_+ (characteristic timescale for ion motion)

Density Scaling

• n* = n / λ_0
• p* = p / λ_0
• P* = P / N_0
• Where λ_0 = F_ph * b (characteristic density from photon flux)

Potential Scaling

• φ* = φ / V_T
• Where V_T = thermal voltage

Key Length Scale

• Debye length: L_d = sqrt(ε * V_T / (2 * q^2 * N_0))

Key Variables

• V_ap: Applied voltage
• V_bi: Built-in potential
• R_l, R_r: Surface recombination rates (left/right interfaces)
• τ_ion: Characteristic timescale for ion motion
• L_d: Debye length

Output

A complete set of constraints defining the system edges and normalized variables for numerical solution.