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T6 · Mechanics / StrengthPolymer Nanotechnology & NanocompositesGATE 2027 Syllabus (XE5)
Electrical Percolation Threshold in Carbon Nanotube Nanocomposites
Power-law scaling equation describing the sharp insulator-to-conductor transition in polymer nanocomposites filled with conductive nanoparticles (carbon nanotubes, graphene, or carbon black).
Mathematical Canonical Relation
Plain Formula: sigma = sigma_0 * (phi - phi_c)^t
Live Engineering Parameter Solver
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phi_vol
Range: 0.002 – 0.1 vol fraction
phi_c_val
Range: 0.001 – 0.05 vol fraction
sigma0_val
Range: 500 – 50000 S/m
exp_t
Range: 1.4 – 2.5 dimensionless
Calculated Engineering Result:
sigma = 2.89 S/m
Engineering Verification:Electrically conductive percolation network formed.
When & Where to Apply
Use when formulating electrically conductive plastics for Electrostatic Discharge (ESD) protection, EMI shielding, and fuel tank grounding.
Common Engineering Pitfalls
- Assuming spherical particle percolation threshold (approx 16 vol%) applies to high aspect ratio CNTs (which percolate at <0.5 vol%)
- Applying the equation below the percolation threshold phi < phi_c
Physical Assumptions & Validity Limits
- Volume fraction phi is above the critical percolation threshold phi_c
- Uniform dispersion without severe agglomeration
- Universal critical exponent t approx 1.6 to 2.0 for three-dimensional random networks
Variable Symbols, SI Units & Dimensional Specifications
| Symbol | Variable Meaning | SI Unit | Typical Industrial Range |
|---|---|---|---|
| \sigma | Electrical conductivity of nanocomposite | S/m | — |
| \sigma_0 | Conductivity scaling factor (filler intrinsic conductivity) | S/m | 10000 |
| \phi | Volume fraction of conductive nanofiller | volume fraction | 0.025 |
| \phi_c | Critical percolation threshold volume fraction | volume fraction | 0.008 |
| t | Critical universal conductivity exponent | dimensionless | 2.0 |
Worked Industrial Numerical Example: Multi-Walled Carbon Nanotube (MWCNT) / Polycarbonate ESD Compound
Problem Statement:
MWCNTs dispersed in Polycarbonate have percolation threshold phi_c = 0.008 (0.8 vol%). Scaling factor sigma0 = 1.0 × 10^4 S/m, exponent t = 2.0. Calculate conductivity at filler loading phi = 0.025 (2.5 vol%).
Step-by-Step Numerical Solution:
- Calculate effective volume above threshold: phi - phi_c = 0.025 - 0.008 = 0.017
- Raise to power t = 2.0: (0.017)^2 = 2.89 × 10^-4
- Multiply by sigma0: 10,000 * 2.89 × 10^-4 = 2.89 S/m
Final Answer: sigma = 2.89 S/m (Volume Resistivity rho = 0.35 Ohm·m)
💡 Engineering Interpretation: The conductivity jumps by over 14 orders of magnitude from base PC (10^-14 S/m) to 2.89 S/m, placing the compound firmly in the ESD and EMI shielding protection range.
Authoritative Technical Reference & Source Provenance
Electrical Conductivity of Carbon Nanotube-Polymer Composites by W. Bauhofer & J.Z. Kovacs (Composites Science and Technology, 2009), Vol 69, Page 1486-1498.
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