An elegant interplay of engineering precision and scientific control. This is a perfect entry into statistical modeling and control in nanoimprint lithography or microfabrication.
π Top-Level Overview
Top Panel (Cross-section & DoE Pattern)
- Si stamp with cavities and protrusions presses into a PMMA polymer.
- The patterned grid suggests a Design of Experiments (DoE) layout.
Bottom Panel (3D Topography)
- Likely obtained from AFM or profilometry.
- Height data shows feature fidelity and pattern replication.
π Statistical Modeling & Process Control
π§ 1. Process Characterization
Use Design of Experiments to vary parameters like temperature, pressure, and time. Model outputs such as fidelity, roughness, and height.
π 2. Response Surface Methodology (RSM)
Y = Ξ²₀ + Σββ Xβ + Σβββ Xβ² + Σβββ Xβ Xβ + Ξ΅
Where Y is the output, Xi are input factors, and Ξ΅ is error.
π 3. Statistical Process Control (SPC)
Use control charts to monitor process drift or anomalies (e.g., ̄X and R-charts).
π§ 4. Machine Learning
- PCA for dimension reduction
- Random Forests / SVM for defect classification
- Gaussian Processes for complex predictions
π Summary
- Physical process ✓
- Statistical design ✓
- Modeled equations ✓
- SPC tools ✓
- Machine learning ✓
π§ Step 1: Define the Process
We model the PMMA embossing process. The response variable is feature height (ΞΌm).
| Variable | Symbol | Units | Levels |
|---|---|---|---|
| Embossing Pressure | X1 | MPa | 5, 10, 15 |
| Embossing Temperature | X2 | °C | 120, 140, 160 |
| Holding Time | X3 | sec | 30, 60, 90 |
π Step 2: Experimental Design
We apply a Central Composite Design (CCD), reducing the 27 runs to 19 using factorial, axial, and center points.
π Step 3: Simulated Data
Y = 0.5 + 0.05X1 + 0.03X2 + 0.02X3 - 0.0005X1X2 + 0.0003X1² + Ξ΅
Ξ΅ ~ N(0, 0.05²)
Here are the first 10 simulated data points:
| Run | Pressure (MPa) | Temp (°C) | Time (s) | Height (ΞΌm) |
|---|---|---|---|---|
| 1 | 5.00 | 120.0 | 30.0 | 4.68 |
| 2 | 5.00 | 120.0 | 90.0 | 5.85 |
| 3 | 5.00 | 160.0 | 30.0 | 5.79 |
| 4 | 5.00 | 160.0 | 90.0 | 7.03 |
| 5 | 15.00 | 120.0 | 30.0 | 4.61 |
| 6 | 15.00 | 120.0 | 90.0 | 5.81 |
| 7 | 15.00 | 160.0 | 30.0 | 5.60 |
| 8 | 15.00 | 160.0 | 90.0 | 6.76 |
| 9 | 18.41 | 140.0 | 60.0 | 5.61 |
| 10 | 1.59 | 140.0 | 60.0 | 5.90 |
π Regression Model
Height = 0.529 + 0.022X1 + 0.030X2 + 0.020X3 - 0.0004X1X2 + 0.0012X1²
R² = 0.997 → nearly perfect fit.
Significant Terms: Temperature, Time, Interaction (X1*X2), Pressure²
Would you like to visualize the surface, monitor with SPC charts, or export the dataset and code?
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