Author: [Bruce Zhou]
Affiliation: [Jota Machinery Composites Material Prepreg Solution]
Corresponding Author: [jotamachinery@gmail.com]
Published : December 08 , 2025
Abstract
Accurate thermal characterization of carbon-fiber/epoxy (CF/EP) composites is essential for predicting temperature evolution in filament-wound natural-gas tanks during rapid filling. Conventional 3ω methods were developed primarily for isotropic thin films and do not directly address bulk, strongly anisotropic laminates. This work consolidates and interprets a 179-page PhD study that extends the 3ω technique to bulk orthotropic CF/EP using a two-dimensional anisotropic heat-conduction model and inverse parameter estimation.
A platinum line heater/sensor, deposited on the composite surface and driven at angular frequency ω, generates a 2ω surface temperature oscillation that is inferred from the 3ω voltage component of a Wheatstone bridge. A Green’s-function–based forward model for anisotropic heat conduction predicts the complex temperature response; least-squares fitting of amplitude and phase yields the in-plane and through-thickness conductivities (kx, ky) and effective diffusivity. The method is first validated on isotropic polymethyl methacrylate (PMMA), where extracted diffusivities are within ~0.6% of handbook values and show excellent repeatability across samples and boundary conditions. It is then applied to unidirectional CF/EP laminates relevant to natural-gas tanks.
Measured in-plane conductivities lie in the range 5.28–7.58 W·m⁻¹·K⁻¹, while through-thickness conductivities are 0.52–0.72 W·m⁻¹·K⁻¹, giving an average anisotropy ratio kx/ky ≈ 10.3, consistent with reported literature values for similar composites. Effective thermal diffusivities fall around 4.6×10⁻⁶ m²·s⁻¹. Sensitivity analysis identifies frequency windows where phase is distinctly sensitive to kx and ky, allowing simultaneous estimation of both components. The extended 3ω approach therefore provides a compact, frequency-domain means of characterizing bulk anisotropic composites, with direct relevance to transient heat-transfer modelling in high-pressure composite tanks. Practical challenges remain in metal-film deposition on rough composite surfaces and in managing long acquisition times, but the methodology is broadly transferable to other orthotropic composite systems.
Keywords
3ω method; anisotropic thermal conductivity; carbon-fiber/epoxy composites; natural-gas tanks; Green’s functions; thermal diffusivity; in-plane and through-thickness conduction; inverse parameter estimation
1. Introduction
Carbon-fiber/epoxy composites are widely used for high-pressure natural-gas and hydrogen storage tanks due to their high specific strength and stiffness. During rapid filling, compressed gas heats the inner liner and surrounding laminate, leading to temperature gradients that can influence allowable fill rates, liner integrity and long-term durability. Predictive models for such tanks require reliable orthotropic thermal properties: in particular, the thermal conductivities along the fiber direction and through the thickness.
While constituent-based models can estimate effective conductivities from fiber and matrix data, volume fractions and idealized microstructures, their accuracy is limited by simplifying assumptions and uncertainty in constituent properties. Direct measurements on real laminates are therefore attractive but challenging, especially when properties are direction-dependent and the material is relatively thick.
The steady-periodic 3ω method introduced by Cahill and co-workers for thin films offers high sensitivity to thermal transport, but its original formulation assumes isotropy and simple one-dimensional heat flow. Tian’s 2011 dissertation extends this technique to bulk anisotropic CF/EP composites by combining precise electrical measurement with a two-dimensional anisotropic heat-conduction model and inverse parameter estimation. The work is motivated by the need to characterize the composite overwrap of natural-gas tanks, but the methodology has broader implications for composite thermal metrology.
2. The 3ω Method for Bulk Anisotropic Materials
2.1 Principle of the 3ω technique
In the 3ω method, a narrow metal line deposited on a sample surface acts simultaneously as heater and thermometer. An AC current at angular frequency ω flows through the line, generating Joule heating at frequency 2ω. The line’s resistance depends on temperature; hence its resistance oscillates at 2ω, and the voltage across it contains a component at 3ω. The magnitude and phase of this 3ω voltage encode the amplitude and phase of the underlying 2ω temperature oscillation at the sample surface.
For an idealized geometry, the temperature oscillation can be solved analytically or numerically as a function of material properties, geometry, frequency and boundary conditions. By fitting the model to the measured response, one can extract thermal properties such as conductivity and diffusivity.
2.2 Extension to anisotropic bulk composites
When the substrate is a bulk orthotropic composite rather than an isotropic semi-infinite solid, heat spreads differently along the fiber direction and through the thickness. Tian formulates a two-dimensional anisotropic heat-conduction model for a rectangular CF/EP plate with a surface line heater, including convective losses at the exposed boundaries. A coordinate transformation converts the orthotropic conduction equation into an equivalent isotropic form, allowing the use of Green’s functions for harmonic heating. The model predicts the complex temperature response at the heater averaged over its width.
Importantly, both in-plane and through-thickness conductivities enter the solution, and their influence on the phase and amplitude is frequency-dependent. This creates the possibility of identifying kx and ky simultaneously by analyzing the frequency dependence of the 3ω signal.
3. Experimental Implementation
3.1 Sample preparation and surface conditioning
Two types of specimens are prepared:
- PMMA reference samples for isotropic validation.
- Carbon-fiber/epoxy plates cut from a filament-wound tank material.
Composite specimens are typically on the order of 1″ × 1″ × 3 mm. Surface preparation is critical, particularly for CF/EP, where roughness and exposed carbon fibers complicate metal film deposition:
- The composite surface is progressively polished using alumina suspensions (3 μm down to 0.05 μm) to reduce roughness and remove gross defects.
- A thin epoxy isolation layer is brushed onto the polished surface to electrically insulate the heater from the conductive carbon reinforcement and to provide a more uniform substrate.
- The epoxy layer is re-polished to achieve a reasonably smooth, continuous surface suitable for metal deposition.
3.2 Heater deposition and electrical setup
A platinum film of nominal thickness ~50 nm and width on the order of 100 μm is sputter-deposited onto the prepared surface, forming the 3ω heater/sensor. The line is patterned such that its length exceeds its width by at least an order of magnitude, approximating an infinite line for thermal modelling.
Electrical measurements are performed using a Wheatstone bridge configuration:
- The Pt heater forms one arm of the bridge, with three precision resistors completing the circuit.
- An AC excitation is applied, and the 3ω component of the bridge output is measured using a lock-in amplifier.
- Careful calibration with known resistors replaces the heater to quantify parasitic capacitances and resistances in the cables and instrumentation.
- An impedance model converts the measured 3ω voltage into an effective 2ω temperature oscillation of the heater, which becomes the input to the thermal inverse problem.
Frequency sweeps typically span several orders of magnitude (e.g. 0.001–10,000 Hz), and multiple excitation levels are used to assess linearity and optimize signal-to-noise without overheating the sample.
4. Thermal Forward Model and Parameter Estimation
4.1 Anisotropic Green’s-function model
The two-dimensional anisotropic heat equation is solved in the frequency domain using Green’s functions. For a given heater geometry, sample thickness, and boundary conditions (including convection at exposed surfaces), the model predicts the complex temperature response beneath the heater for each frequency.
Key elements include:
- Coordinate transformation to handle orthotropic conductivities (kx along the fiber plane, ky through the thickness).
- Integration over the heater width to obtain an average temperature relevant to the electrical measurement.
- Inclusion of sample dimensions and convective heat transfer, enabling realistic modelling of finite plates rather than semi-infinite solids.
4.2 Sensitivity analysis
To ensure that kx and ky can be uniquely determined, a sensitivity analysis is performed. The sensitivity of the phase of the heater temperature to kx and ky is evaluated across frequency, revealing distinct regimes:
- At higher frequencies (≈50–500 Hz), the phase is more sensitive to in-plane conductivity kx, because thermal waves penetrate less deeply and heat flow is dominated by lateral spreading.
- At lower frequencies (≈0.01–0.1 Hz), the phase is more sensitive to through-thickness conductivity ky, since thermal penetration depth increases and vertical heat flow becomes significant.
This separation of sensitivity provides a basis for simultaneous estimation of kx and ky from a single heater under a frequency sweep.
4.3 Least-squares inverse parameter estimation
Thermal properties are extracted by minimizing a weighted least-squares cost function that compares model predictions to measured data:
- Both amplitude and phase of the temperature oscillation are used, improving robustness compared to phase-only or amplitude-only approaches.
- For PMMA, an isotropic model is used and sensitivity focuses on thermal diffusivity α.
- For CF/EP, the anisotropic model is used, with kx, ky and α as adjustable parameters.
The optimization yields best-fit values and provides a basis for estimating confidence intervals and assessing parameter correlations.
5. Validation with PMMA
Before applying the method to composites, PMMA is used as a reference isotropic material with well-tabulated thermal diffusivity. Multiple specimens and experimental configurations are tested:
- Samples P#6, P#8, P#10 and P#14 are measured, often under different boundary materials (e.g. aluminum, wood, air) to vary convective conditions.
- Extracted diffusivities cluster tightly around α ≈ 1.284×10⁻⁷ m²·s⁻¹.
The table below summarizes representative results compared with the handbook value αₕ = 1.276×10⁻⁷ m²·s⁻¹:
| Sample | Experiment | α (m²·s⁻¹) | % Error vs αₕ |
|---|---|---|---|
| P#6 | E#1 | 1.235×10⁻⁷ | −3.2% |
| E#2 | 1.242×10⁻⁷ | −2.7% | |
| P#8 | E#1 | 1.284×10⁻⁷ | +0.6% |
| E#2 | 1.285×10⁻⁷ | +0.7% | |
| P#10 | E#1 | 1.284×10⁻⁷ | +0.6% |
| E#2 | 1.285×10⁻⁷ | +0.7% | |
| P#14 | E#1 | 1.285×10⁻⁷ | +0.7% |
| E#2 | 1.285×10⁻⁷ | +0.7% | |
| E#3 | 1.285×10⁻⁷ | +0.7% |
For the best-behaved specimens, average error is on the order of 0.6%, and repeated experiments yield essentially identical diffusivities. Extracted Biot numbers and effective convection coefficients for different boundary materials also align with expectations, lending further confidence to the combined experimental–numerical approach.
6. Thermal Properties of Carbon-Fiber/Epoxy Composites
The extended 3ω method is then applied to carbon-fiber/epoxy composite specimens cut from a natural-gas tank laminate. Four representative samples (C#7, C#8, C#13, C#16) are analyzed using the anisotropic forward model and the same inverse procedure.
The extracted in-plane and through-thickness conductivities and diffusivities are summarized below:
| Sample | kx (W·m⁻¹·K⁻¹) | ky (W·m⁻¹·K⁻¹) | α (×10⁻⁶ m²·s⁻¹) | Phase Offset (°) |
|---|---|---|---|---|
| C#7 | 5.28 | 0.52 | 3.75 | −0.33 |
| C#8 | 5.28 | 0.72 | 3.81 | −0.35 |
| C#13 | 7.16 | 0.52 | 5.16 | −0.35 |
| C#16 | 7.58 | 0.69 | 5.46 | −0.38 |
| Avg | 6.32 | 0.61 | 4.55 | −0.35 |
Several observations follow:
- Strong anisotropy: The average ratio kx/ky ≈ 10.3, indicating that heat flows more than ten times more readily along the fiber plane than through the thickness. This ratio is consistent with literature values (≈9.8–11.5) for similar CFRP systems.
- Magnitude of conductivities: In-plane conductivities between 5–8 W·m⁻¹·K⁻¹ and through-thickness conductivities between 0.5–0.7 W·m⁻¹·K⁻¹ are physically reasonable for unidirectional CF/EP laminates.
- Diffusivity: Effective diffusivities in the range (3.7–5.5)×10⁻⁶ m²·s⁻¹ reflect both conduction and volumetric heat capacity of the composite.
Variation among specimens is attributed primarily to microstructural differences beneath the heater—specifically local fiber orientation and resin-rich regions. This local microstructure influences the effective thermal pathway accessed by the line heater.
7. Discussion
7.1 Relevance to natural-gas tank design
For filament-wound natural-gas tanks, the anisotropic thermal properties measured here have direct implications:
- Rapid filling: As gas enters the tank, heat generated by compression must be conducted through the liner and composite overwrap. High in-plane conductivity promotes lateral heat spreading around the circumference, while relatively low through-thickness conductivity slows heat transfer toward the outer surface.
- Temperature gradients: Anisotropy can lead to non-uniform temperature fields, with local hot spots influenced by winding angle, layer stacking sequence and local fiber architecture.
- Model calibration: The measured values of kx and ky provide realistic parameters for finite-element thermal models of tanks, improving predictions of peak temperature and temporal evolution during filling protocols.
7.2 Strengths of the extended 3ω approach
The combined experimental–numerical framework offers several attractive features:
- Simultaneous estimation of kx and ky using a single surface heater and a frequency sweep, rather than multiple specimen orientations or separate experiments.
- Frequency-domain operation, which naturally filters some environmental noise and avoids the need for step-change heat inputs and direct heat-flux measurement.
- Validated forward model, supported by the close agreement with PMMA reference data and physically reasonable Biot number estimates.
7.3 Practical challenges and limitations
Several practical issues and methodological limitations are also evident:
- Metal film deposition on composites: Achieving a continuous, low-defect Pt film on rough composite surfaces is non-trivial. Many experiments fail due to film cracking, poor adhesion or electrical discontinuities. This limits throughput and introduces selection bias toward smoother regions.
- Measurement time: Full frequency sweeps at multiple excitation levels can require tens of hours per configuration, which is acceptable for research but may be impractical for industrial quality control without substantial automation and parallelization.
- Model simplifications: The forward model neglects heater heat capacity and thermal boundary resistance between film and substrate. These approximations are acceptable at lower frequencies but may distort high-frequency responses, potentially biasing estimates of in-plane properties.
- Limited statistical sample: Only a small number of CF/EP specimens are analyzed, so the full range of manufacturing variability is not captured.
7.4 Opportunities for future work
Future developments could address these limitations and extend the methodology:
- Improved surface preparation and deposition—for example, using sputtered adhesion layers, optimized epoxy isolation films or alternative metallization strategies for rough composites.
- Vacuum measurements to further suppress convective losses and environmental noise, extending usable frequency windows.
- Inclusion of heater heat capacity and interface resistance in the forward model to improve accuracy at higher frequencies.
- Orientation-resolved measurements, with heaters aligned along different in-plane directions relative to fiber orientation, to probe off-axis conductivity in multi-directional laminates.
- Application to other composite systems, such as fabric-reinforced laminates, hybrid fiber architectures, or porous insulators where anisotropy is strong.
8. Conclusions
An extended 3ω method combining careful experimentation, an anisotropic two-dimensional heat-conduction model and inverse parameter estimation provides a practical route to measure thermal properties of bulk carbon-fiber/epoxy composites. Validation against PMMA reference materials yields diffusivities within about 0.6% of handbook values, confirming the reliability of the system.
Applied to CF/EP laminates from natural-gas tank structures, the technique delivers in-plane conductivities around 6.3 W·m⁻¹·K⁻¹ and through-thickness conductivities around 0.6 W·m⁻¹·K⁻¹, with an anisotropy ratio kx/ky ≈ 10.3. These values are consistent with expectations from composite theory and literature measurements and provide essential inputs for transient heat-transfer modelling in high-pressure composite vessels.
Although challenges remain in film deposition, long measurement times and model refinements, the extended 3ω approach constitutes a robust framework for anisotropic thermal characterization of composites. With further development, it is well positioned to support the design and assessment of advanced composite structures in energy storage, aerospace and transportation applications.