Design Rules for
Micro-Fabricated Piezoresistive Sensors

Project Overview

This research establishes design rules for highly sensitive piezoresistive pressure sensors using finite element analysis (FEA) in COMSOL Multiphysics. The study addresses a critical gap in the literature: while many micro-patterned piezoresistive sensors have been developed, there are no systematic design optimization guidelines. The work provides quantitative recommendations for micro-feature shape, material conductivity, spatial density, feature size, and pyramid angle to maximize sensitivity and current output for wearable pulse wave monitoring applications.

Problem Addressed

• Most piezoresistive sensors output nanoampere-range currents, requiring bulky sourcemeters for signal acquisition
• No design rules exist for optimizing micro-feature geometry in piezoresistive sensors
• Trade-off between sensitivity and linearity not systematically studied
• Need for milliamp-range output to enable truly wearable signal acquisition with simple electronics

Simulation Methodology

COMSOL Multiphysics Setup:

• Physics Interfaces: Electric Currents (ec) + Solid Mechanics module
• Formulation: Arbitrary Lagrangian-Eulerian (ALE) method for large deformation problems
• Contact Algorithm: Augmented Lagrangian method (more accurate than default penalty method)
• Mesh Type: Free tetrahedral with "extremely fine" size calibration
• Convergence Criteria: DOF increased to 1000% until results varied <5%

Simulation Parameters:

Results 1: Micro-Feature Shape Comparison

Ranking by Sensitivity (0-12 kPa range):

1. Micro-dome - Highest sensitivity (but non-linear response)
2. Micro-pyramid - Excellent sensitivity + linear response (best overall)
3. Micro-cone - Moderate sensitivity
4. Micro-pillar - Lowest sensitivity (good for digital switching applications)

Analytical Derivation of Contact Area:

Using Thales theorem, the contact area for each shape was derived:

Pyramid contact area: Sₚ = 1.3333 × d²

Cone contact area: S꜀ = 1.0472 × d²

Dome contact area: Sᴅ = π × (√(100d - d²))²

Where d is the displacement (layer spacing decrease). These analytical equations match COMSOL simulation results, validating the model.

Results 2: Effect of Material Conductivity

Key Findings:

• Sensitivity saturation: Sensor sensitivity saturates at elastomeric layer conductivity of 10 S/m
• Below 1 S/m: Sensitivity deteriorates significantly (typical of CNT-PDMS composites)
• Gold coating (10⁵ S/m): Current output in milliamp range (vs. nanoamp range for conductive polymers)
• Higher current = Higher SNR = Simple electronics (Arduino, ESP32) instead of bulky sourcemeters

Conductivity Recommendations:

Results 3: Geometric Parameter Optimization (Micro-Pyramid)

A. Spatial Number Density:

• Low density (3 pyramids/mm): Highest sensitivity (pressure concentrated at fewer contact points)
• Medium density (4.5 pyramids/mm): Moderate sensitivity
• High density (7 pyramids/mm): Lowest sensitivity (pressure distributed across more pyramids)
• Optimal recommendation: 3 pyramids per millimeter

B. Pyramid Angle (α):

• Optimal range: 50° - 60°
• Balance between contact area growth rate and localized pressure
• Angles <50°: High contact area growth but low localized pressure
• Angles >60°: High localized pressure but low contact area growth

C. Feature Base Size:

• Optimal size: 100 μm
• Smaller features → higher localized pressure → higher sensitivity
• Tested sizes: 100 μm, 150 μm, 200 μm (all with 60° angle, low density)

Summary of Design Rules

Governing Equations

Charge Conservation (Continuity Equation):

∇·J = Qⱼᵥ

Current Density:

J = σE + Jₑ

Electric Field Strength:

E = -∇V

Where: J = current density, σ = electrical conductivity, E = electric field strength, V = electrical potential, Jₑ = externally generated current density.

Practical Implications for Wearable Sensors

Current Output Comparison:

• Conductive polymer sensors (CNT-PDMS): ~nanoamp output → requires bulky sourcemeter
• Gold-coated sensors (this design): ~milliamp output → readable by Arduino/ESP32

Enabling Truly Wearable Systems:

• High current output eliminates need for expensive, desktop-sized sourcemeters
• Simple voltage divider circuits + development boards (Arduino Nano, ESP32) sufficient
• Higher signal-to-noise ratio enables reliable pulse waveform detection
• Battery-powered operation possible (1000 mAh LiPo)

Software & Tools Used

• COMSOL Multiphysics (FEA)
• Electric Currents (ec) Interface
• Solid Mechanics Module
• Arbitrary Lagrangian-Eulerian (ALE) Method
• Augmented Lagrangian Contact Algorithm
• Free Tetrahedral Meshing

Downloads

• Full Journal Paper (PDF)

My Contributions

• Validation of simulation results against experimental data
• Data analysis and interpretation
• Manuscript writing, review, and editing

Funding

• National Science Foundation (NSF) PATHS-UP ERC (Award #1648451)
• NSF Awards #2126190 and #2107318

Related Publications

• Jafarizadeh, B., Chowdhury, A.H., Khakpour, I., Pala, N., Wang, C. "Design Rules for a Wearable Micro-Fabricated Piezo-Resistive Pressure Sensor." Micromachines, 2022, 13, 838.
• Jafarizadeh, B., Chowdhury, A.H., Sozal, M.S.I., Cheng, Z., Pala, N., Wang, C. "Hardware and Protocol for Testing of Piezoresistive Pressure Sensors for Pulse Wave Monitoring." IEEE Sensors Letters, 2023, 7(8), 2502304.
• Jafarizadeh, B., Chowdhury, A.H., Sozal, M.S.I., Cheng, Z., Pala, N., Wang, C. "Wearable System Integrating Dual Piezoresistive and Photoplethysmography Sensors for Simultaneous Pulse Wave Monitoring." ACS Applied Materials & Interfaces, 2024, 16, 65402-65413.

Conclusion

This study established the first comprehensive design rules for micro-fabricated piezoresistive pressure sensors using finite element analysis. Key findings include: (1) micro-domes and micro-pyramids offer highest sensitivity, with pyramids providing better linearity; (2) material conductivity must exceed 10 S/m (preferably gold coating at 10⁵ S/m) to achieve milliamp current output for wearable signal acquisition; (3) optimal geometric parameters are 3 features/mm spatial density, 100 μm feature size, and 50-60° pyramid angle. These design rules enable the development of truly wearable pressure sensors that can be read by simple, low-cost electronics (Arduino, ESP32) instead of bulky sourcemeters, advancing the field toward practical, continuous cardiovascular monitoring.