The Thermoelastic Component of the Photoacoustic Response in a 3D-Printed Polyamide Coated with Pigment Dye: A Two-Layer Model Incorporating Fractional Heat Conduction Theories
Abstract
1. Introduction
2. Theory
2.1. Fractional Thermal Diffusion
2.2. Temperature Variations
- One-dimensional spatial approximation: The sample is uniformly illuminated over its surface, and the lateral dimensions are significantly larger than the thermal diffusion length in the modulation frequency range. This justifies the use of a one-dimensional geometry along the axis perpendicular to the surface, as commonly employed in photoacoustic modeling [8,9,15,70].
- Sinusoidal optical modulation and lock-in detection: The excitation source is modulated sinusoidally in time, and the resulting acoustic signal is recorded using a lock-in amplifier. Consequently, all physical quantities of interest—such as temperature, displacement, and pressure—are represented in the frequency domain using their spectral counterparts.
- Volumetric optical absorption in both layers, as opposed to the more commonly assumed surface absorption scenario [17,71,72,73,74]. This extension is crucial for describing polymeric or composite materials with structural disorder. The absorbed irradiance is calculated using Beer–Lambert’s law of absorption, , where I0 is the incident radiation’s peak intensity in [W/m2], is the optical absorption coefficient in [1/m], and is the optical reflection coefficient.
- The absence of absorption and heat conduction in the surrounding gas—Air is regarded as completely transparent, meaning it does not absorb incoming optical radiation and, as a result, does not generate heat sources. This property enables all incoming radiation to reach the first layer. Additionally, air is known to be a poor conductor of heat, which is why we apply adiabatic boundary conditions [18,69].
- The incorporation of anomalous diffusion effects via generalized Cattaneo equation (GCE) models—specifically GCEI, GCEII, and GCEIII—which allow for structural heterogeneity and memory effects in heat transport, both of which are relevant in polymeric systems with high entropy and fractal structures [17,37,39,40,64,76].
2.3. Thermoelastic Bending Effect
- It is considered that the circular plate is simply supported.
- A two-layer model utilizing generalized heat conduction theories with an infinite optical absorption coefficient for the first layer [17]—corresponding to in our model. This model was developed to describe optically opaque bilayer structures, which commonly arise in photoacoustic investigations of materials with high reflectivity. In such cases, a thin coating is applied to the sample surface to reduce optical reflection, and the sample is oriented during measurement so that the coating is illuminated. This coating typically has a very high optical absorption coefficient, ensuring that the entire incident electromagnetic energy is absorbed within a very thin surface layer—effectively at the very surface of the sample—since the optical absorption parameter .
- A two-layer model based on classical heat conduction theory for opaque layers [16]—aligning with and in our model. Similarly to the previous case, this model was developed for optically opaque bilayer samples, in which the illuminated layer exhibits a very high optical absorption coefficient, such that . However, in contrast to the previously described model, the classical model A Priori neglects the possible presence of anomalous diffusion effects and the finite speed of heat propagation.
- A single-layer model from hyperbolic heat conduction theory, that incorporates volumetric absorption of incident radiation and a coating on the unilluminated side (the microphone side) [22]—corresponding to and in the derived model. This model was developed to enable photoacoustic measurements of optically transparent or semi-transparent samples in a way that allows the determination of the sample’s optical absorption coefficient, regardless of how low it may be. In such cases, a thin coating with a very high optical absorption coefficient is applied to the transparent or semi-transparent sample, and the measurement configuration is such that this highly absorbing coating is illuminated. In this model, the possible presence of anomalous thermal diffusion in the optically transparent layer is neglected, as is the contribution of heat conduction through the coating, i.e., the coating is considered solely as an optical layer.
3. Results
3.1. Analysis of Anomalous Diffusion Effects
3.2. Thermoelastic Component of Photoacoustic Signal for PA12 Coated Sample
3.2.1. The Influence of the Order of the Fractional Derivative of a Transparent PA12 Sample and the Coating Thickness on the Thermoelastic Component
3.2.2. The Influence of the Order of the Fractional Derivative of a Whitish PA12 Sample and the Coating Thickness on the Thermoelastic Component
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
GCE | Generalized Cattaneo Equation |
3D- | Three-Dimensional |
FDM | Fused Deposition Modeling |
SLA | Stereolithography |
SLS | Selective Laser Sintering |
TE | Thermoelastic |
CTRW | Continuous-Time Random Walk |
CHT | Classical Heat Theory |
HHT | Hyperbolic Heat Theory |
1LM | One-Layer Model |
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Popovic, M.N.; Galovic, S.P.; Lenzi, E.K.; Somer, A. The Thermoelastic Component of the Photoacoustic Response in a 3D-Printed Polyamide Coated with Pigment Dye: A Two-Layer Model Incorporating Fractional Heat Conduction Theories. Fractal Fract. 2025, 9, 456. https://doi.org/10.3390/fractalfract9070456
Popovic MN, Galovic SP, Lenzi EK, Somer A. The Thermoelastic Component of the Photoacoustic Response in a 3D-Printed Polyamide Coated with Pigment Dye: A Two-Layer Model Incorporating Fractional Heat Conduction Theories. Fractal and Fractional. 2025; 9(7):456. https://doi.org/10.3390/fractalfract9070456
Chicago/Turabian StylePopovic, Marica N., Slobodanka P. Galovic, Ervin K. Lenzi, and Aloisi Somer. 2025. "The Thermoelastic Component of the Photoacoustic Response in a 3D-Printed Polyamide Coated with Pigment Dye: A Two-Layer Model Incorporating Fractional Heat Conduction Theories" Fractal and Fractional 9, no. 7: 456. https://doi.org/10.3390/fractalfract9070456
APA StylePopovic, M. N., Galovic, S. P., Lenzi, E. K., & Somer, A. (2025). The Thermoelastic Component of the Photoacoustic Response in a 3D-Printed Polyamide Coated with Pigment Dye: A Two-Layer Model Incorporating Fractional Heat Conduction Theories. Fractal and Fractional, 9(7), 456. https://doi.org/10.3390/fractalfract9070456