Temperature Effect on Crystal Growth
Reference ID: MET-129F | Process Engineering Reference Sheets Calculation Guide
The optimal temperature window depends on the crystal orientation and dopant levels, but generally:
- Seed zone: 1 450 °C ± 5 °C to maintain a stable solid‑liquid interface.
- Growth zone: 1 420 °C – 1 440 °C for most Czochralski (CZ) processes.
- Crucible wall: keep below 1 500 °C to avoid crucible degradation.
Maintaining these temperatures within ±2 °C reduces dislocation density and impurity segregation.
A steep axial temperature gradient (≥ 15 °C/cm) can:
- Accelerate solidification, increasing thermal stress and generating micro‑cracks.
- Promote constitutional supercooling, leading to cellular or dendritic structures.
Conversely, a shallow gradient (5 – 10 °C/cm) promotes a planar interface and lowers defect density, but may reduce growth rate.
Follow a controlled thermal profile to avoid thermal shock:
- Ramp‑up: increase temperature at ≤ 10 °C/min until reaching the melt temperature.
- Stabilize: hold for 30–60 min to achieve thermal equilibrium.
- Ramp‑down: cool at 2 – 5 °C/min until the crystal reaches the annealing temperature, then reduce to ≤ 1 °C/min for final cool‑down.
Monitoring thermocouple drift and adjusting rates in real time improves crystal uniformity.
Yes. Fluctuations of ±3 °C or more can cause:
- Transient changes in segregation coefficient, leading to dopant spikes.
- Variations in melt convection, which redistribute dopants unevenly.
To maintain uniform dopant profiles, implement:
- Closed‑loop temperature control with ±0.5 °C tolerance.
- Real‑time dopant monitoring (e.g., laser‑induced breakdown spectroscopy).
Worked Example – Estimating Growth Rate of Sodium Chlorate Crystals at 25 °C
A continuous crystallizer is being commissioned to produce sodium chlorate monohydrate. The design team needs to know the linear growth rate of the crystals at the operating temperature of 25 °C so that the residence time can be fixed. Laboratory data give an empirical rate constant, but the team also wants to check the value against the Arrhenius model to confirm reliability.
Knowns- Universal gas constant, R = 8.314 J mol⁻¹ K⁻¹
- Pre-exponential factor, A = 1.0 × 10⁵ m s⁻¹
- Activation energy, Ea = 50 000 J mol⁻¹
- Temperature coefficient, β = 0.02 K⁻¹
- Reference solubility at T0 = 25 °C, S0 = 200 kg m⁻³
- Diffusion coefficient, D = 1.0 × 10⁻⁹ m² s⁻¹
- Characteristic length, L = 0.01 m
- Mass-transfer coefficient, kD = 1.0 × 10⁻⁷ m s⁻¹
- Empirical interface rate constant, ki,emp = 5.0 × 10⁻⁸ m s⁻¹
- Operating temperature, TC = 25 °C (TK = 298.15 K)
- Bulk concentration, C∞ = 220 kg m⁻³
- Supersaturation, σ = 10 % (ΔC = 20 kg m⁻³)
- Compute the Arrhenius interface rate constant: \[ k_{i,Arr} = A\,\exp\!\left(\frac{-E_a}{R\,T_K}\right) = 1.0\times10^{5}\;\exp\!\left(\frac{-50\,000}{8.314\times298.15}\right) = 1.737\times10^{-4}\;\text{m s}^{-1} \]
- Determine the diffusion-limited growth rate: \[ G_d = \frac{2\,D\,\Delta C}{L\,C_{\infty}} = \frac{2\times1.0\times10^{-9}\times20}{0.01\times220} = 1.818\times10^{-6}\;\text{m s}^{-1} \]
- Compute the interface-limited growth rate (Arrhenius): \[ G_{i,Arr} = 2\,k_{i,Arr}\,\frac{\Delta C}{C_{\infty}} = 2\times1.737\times10^{-4}\times\frac{20}{220} = 3.158\times10^{-5}\;\text{m s}^{-1} \]
- Combine resistances to obtain the overall Arrhenius growth rate: \[ \frac{1}{G_{Arr}} = \frac{1}{G_d} + \frac{1}{G_{i,Arr}} \quad\Rightarrow\quad G_{Arr} = 1.998\times10^{-6}\;\text{m s}^{-1} \]
- For comparison, repeat with the empirical ki,emp: \[ G_{i,emp} = 2\,k_{i,emp}\,\frac{\Delta C}{C_{\infty}} = 2\times5.0\times10^{-8}\times\frac{20}{220} = 9.091\times10^{-9}\;\text{m s}^{-1} \] \[ \frac{1}{G_{emp}} = \frac{1}{G_d} + \frac{1}{G_{i,emp}} \quad\Rightarrow\quad G_{emp} = 6.667\times10^{-7}\;\text{m s}^{-1} \]
The predicted linear growth rate of sodium chlorate crystals at 25 °C is
GArr = 2.0 × 10⁻⁶ m s⁻¹ (Arrhenius-based model),
while the empirical fit gives Gemp = 6.7 × 10⁻⁷ m s⁻¹. Use the more conservative empirical value for residence-time design.
"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise."