Next Article in Journal
Implementation of the Stack-CNN Algorithm for Space Debris Detection on FPGA Board
Previous Article in Journal
Establishing Leaf Tissue Nutrient Standards and Documenting Nutrient Disorder Symptomology of Greenhouse-Grown Cilantro (Coriandrum sativum)
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Research on Viscous Dissipation Index Assessment of Polymer Materials Using High-Frequency Focused Ultrasound

College of Metrology Measurement and Instrument, China Jiliang University, Hangzhou 310018, China
*
Author to whom correspondence should be addressed.
Appl. Sci. 2025, 15(17), 9267; https://doi.org/10.3390/app15179267
Submission received: 23 July 2025 / Revised: 19 August 2025 / Accepted: 20 August 2025 / Published: 22 August 2025
(This article belongs to the Section Acoustics and Vibrations)

Abstract

Featured Application

Unlike conventional methods that rely on low-frequency measurements to distinguish the viscosity of different polymer materials, this study proposes a novel approach utilizing high-frequency ultrasound for polymer viscous dissipation index measurement, which demonstrates significant potential.

Abstract

Polymer viscoelasticity is crucial for mechanical performance, but conventional low-frequency methods struggle to isolate viscous loss—a key viscosity indicator. This study introduces a high-frequency ultrasonic method to differentiate the polymer viscous dissipation index by analyzing acoustic phase shifts. We employ ultrasonic phase-shift thermometry to measure localized temperature increases resulting from minute variations in sound velocity during controlled heating. This allows for the quantification of viscous loss, which is then used to distinguish between different polymer formulations. Experimental and simulation results on a series of polyurethane specimens with varying Shore hardness levels demonstrate that decawatt-range (10–20 W) ultrasonic irradiation enables sensitive and precise differentiation. Notably, the Shore A70 polyurethane sample exhibited a significantly higher viscous dissipation index, evidenced by the largest temperature rise (27.5 °C) and the highest proportion of viscous heating to total power dissipation (93.1%) under 17 W acoustic irradiation. While this study focuses on commercially available polymers, the method can be extended to evaluate key performance parameters, such as tensile modulus and glass transition temperature, in polymers fabricated under various processing conditions, thereby offering a powerful tool for material quality assessment.

1. Introduction

The propagation of ultrasonic waves through a medium is governed by fundamental interaction phenomena, including attenuation, scattering, reflection, and transmission [1]. In solids, this behavior is primarily dictated by the elastic modulus and density. However, the material’s viscosity and its temperature dependence also exert a significant influence on wave propagation [2,3]. Temperature, in particular, plays a critical role among external factors. An increase in temperature not only reduces material density but also alters the elastic modulus, which in turn modifies the material’s viscosity and leads to changes in viscous loss due to thermal absorption [4].
For polymeric materials, their unique viscoelasticity-the simultaneous manifestation of viscous fluid characteristics (energy dissipation) and elastic solid behavior (energy storage) serves as a critical distinguishing feature from metallic materials and represents the most effective indicator of mechanical properties [5,6]. Variations in molecular chain structure, glass transition temperature, and Young’s modulus, which stem from differences in manufacturing processes, raw materials, and key processing parameters like pressure and shear, are directly reflected in the material’s viscosity. Consequently, viscosity in polymers is a complex, temperature-and time-dependent property that varies significantly across different materials [7,8].
When a polymer is subjected to continuous ultrasonic irradiation, energy is dissipated as heat due to internal friction, a phenomenon termed viscous loss [9,10,11]. Because different polymers possess distinct viscosities, their viscous loss characteristics vary accordingly. Conventional Fractional-Power-Dissipation (FPD) methods can quantify total ultrasonic energy loss but cannot differentiate between scattering loss and viscous loss [12]. To isolate the viscous loss of different polymer materials from the total loss, real-time temperature monitoring using thermodynamic formulas presents a viable approach.
Traditional temperature measurement techniques, which rely on achieving thermodynamic equilibrium, are typically limited to surface measurements and cannot capture internal viscous heating phenomena [13,14,15]. Current ultrasonic thermometry technologies offer several alternatives. One approach involves analyzing the time-of-flight shift in ultrasonic echo signals before and after heating to establish a correlation with temperature change [16]. Another method utilizes the frequency shift in the propagating wave, which is also temperature-dependent [17]. Still other technique utilizes the attenuation characteristics of ultrasonic waves during propagation in a medium, combining the relationship between the attenuation coefficient and temperature change to achieve non-destructive measurement of temperature [18].
However, none of the aforementioned techniques utilize the relationship between temperature and viscosity. This study employs a phase-shift thermometry method based on transmitted signals. The core principle lies in detecting phase variations in the transmitted signal, which are caused by minute changes in sound velocity during temperature fluctuations. Ultrasonic irradiation induces viscous heating; by measuring the resulting phase shift, we can determine the internal temperature rise. Correlating this temperature change with viscous loss allows for the calculation of its proportion relative to total energy dissipation. This approach has inherent limitations and is only applicable to linear viscoelastic materials—those exhibiting a stress response that follows the Boltzmann superposition principle with respect to strain history under small-strain conditions. It is not applicable to non-Newtonian materials.
In this work, we investigated seven polyurethane samples with similar densities and sound velocities but varying Shore hardness. Each specimen exhibited a distinct thermal response and viscous dissipation profile under focused ultrasound, reflecting their intrinsic viscous dissipation index differences. The proposed methodology primarily focuses on differentiating materials through their viscous dissipation index derived from viscous loss mechanisms, rather than employing rheometric techniques such as slip-aware multigap protocols utilizing steady torsional or Couette flows [19]. The phase-shift measurement methodology demonstrated high sensitivity in characterizing these material-specific variations, enabling precise differentiation. This technique holds potential for medical applications, where non-contact measurements of phase shifts in biological tissues could assess viscosity-related pathological changes, opening a new frontier in medical ultrasonography.

2. Theoretical Background

Under periodic oscillatory loading, viscoelastic materials [20] develop stress responses at the excitation frequency, resulting in energy dissipation through viscous mechanisms [21,22]. For a linear viscoelastic material whose stress response at any given time exhibits a linear superposition relationship with the strain history in accordance with the Boltzmann superposition principle, the complex modulus is defined as the combination of the storage modulus and the loss modulus:
G * = G + i G = σ 0 ε 0 cos δ + i sin δ
where the real part of the complex modulus is the storage modulus   G , which is the energy stored due to the elastic deformation, the imaginary part of the complex modulus is the loss modulus   G , also called viscous loss, which is the energy lost as heat, σ 0   and ε 0 are the amplitude of the stress and strain, respectively, δ is the lag angle of the strain.
For a viscoelastic material vibrating under a sinusoidal strain of frequency f , the heat generation of the micro unit cell per unit of time is given by [23]:
Q = f · σ ( t ) d ε ( t ) = f π ε 0 σ 0 sin δ
By submitting G = σ 0 s i n δ / ε 0 in Equation (1), the average heat production per unit volume Q is directly related to the loss modulus   G of the material. This equation is a bridge between the macroscopic heat production and the microscopic viscosity of the material:
Q = f π G ε 0 2
In the process of acoustic wave propagation, the acoustic field is usually characterized by the acoustic pressure p , mass velocity v and density ρ   changes in the medium, for focused ultrasound acting on the acoustic and temperature fields inside the biological tissues research, at low power, for linear, no flow of the ideal homogeneous medium is usually used to solve the Helmholtz equation, which is the central governing equation for frequency domain analysis in acoustics [24]:
2 P x + k 2 P x = 0
where is the Laplace operator, the wave number k is defined as k = ω / c , ω = 2 π f   for the angular frequency, P ( x ) indicates the distribution of sound pressure in space.
In the actual propagation, the propagation medium is always non-ideal, the energy of the acoustic wave will gradually decay with the distance, that is, the medium absorbs the acoustic wave, at this time, the fluctuation equation in the solid viscoelastic medium is [5]:
k = 1 3 p j k x k + ρ X j = ρ 2 ξ j t 2 ( j = 1 , 2 , 3 )
where x 1 , x 2 , x 3 is the right-angle coordinate system, X 1 , X 2 , X 3 is the applied force along the coordinate axis, ξ 1 , ξ 2 , ξ 3 is the displacement vector, and ρ is the density of the medium.
In terms of heat transfer, the Pennes [25] biological tissue heat transfer model was used to describe the heat transfer based on the fact that tissues are isotropic and homogeneous continuous media:
ρ c T t = k 2 T + c b ω b ( T b T ) + Q
where ρ represents the density of tissue, c is the tissue specific heat capacity, k is the tissue thermal conductivity, T is the tissue temperature, c b represents the specific heat capacity of blood, ω b is the blood perfusion rate, T b is the blood temperature, and Q represents the ultrasonic heating source. The blood perfusion term can be omitted because the study object is for polymer materials, lacks significant blood perfusion, and heat transfer is mainly dominated by heat conduction. By combining the acoustic field with the Pennes bioheat transfer equation, the energy lost in the form of heat during ultrasonic irradiation of the material can be modeled.
When an intermediate layer with thickness D and acoustic impedance   R 2 is embedded in a medium of acoustic impedance R 1 , an acoustic plane-wave P i incident from Medium 1 into Medium 2. Part of the wave is reflected back into Medium 1 called reflected wave ( p 1 r ), and part transmits into the intermediate layer ( p 1 t ). Upon p 1 t reaching the far boundary of the layer, the wave is again partially reflected ( p 2 r ) and partially transmitted into Medium 1 ( p t ) on the opposite side. Denote the pressure amplitudes of the incident, reflected, and transmitted waves as   P i , P 1 r   and P t respectively. the ratio of transmitted wave sound intensity I t to incident wave sound intensity I i is [26]:
t I = I t I i = P t 2 / 2 ρ 1 c 1 P i 2 / 2 ρ 1 c 1 = 4 4 cos 2 k 2 D + R 2 2 + R 1 2 / R 1 R 2 2 sin 2 k 2 D
In the above equation   R 1 = ρ 1 c 1 ,   R 2 = ρ 2 c 2   ,   k 2 = ω / c 2 , the ratio of reflected wave sound intensity to incident wave sound intensity is:
r I = P 1 r 2 / 2 ρ 1 c 1 P i 2 / 2 ρ 1 c 1 = 1 t I
Thus the ratio of the reflected wave sound pressure amplitude to the incident wave sound pressure amplitude is:
P 1 r P i = 1 4 4 cos 2 k 2 D + R 2 2 + R 1 2 / R 1 R 2 2 sin 2 k 2 D
Traditional Fractional-Power-Dissipation [27] by analyzing the ultrasound in the material of the reflected and transmitted signals to measure its energy loss, the energy of the sound waves reflected back from the interface is the echo reduction ER, and the energy transmitted out of the material is the insertion loss IL:
E R = 20 lg P 1 r P i
I L = 20 lg P t P i
When the transmitted sound wave propagates in the material, the energy is lost due to the scattering caused by the non-uniform structure and the heat absorption caused by the viscosity, and the percentage of the loss caused by the scattering and the viscosity is the   η F P D :
η F P D = 1 P t P i 2 P r P i 2

3. Simulation of Temperature Field

To illustrate the methodological basis, a simulation was performed. As the temperature within the polyurethane rises, a non-uniform temperature field is generated, altering the local speed of sound and inducing a phase shift in the transmitted signal. A coupled acoustic-thermal model was implemented in COMSOL Multiphysics 6.3 using a two-dimensional axisymmetric configuration. The study comprises two distinct analytical approaches: pressure acoustics analysis in the frequency domain, and bioheat transfer analysis in the time domain.
The model was initialized at a uniform temperature of 300 K. A single-element focusing transducer was simulated with an operating frequency of 600 kHz, a focal length of 90 mm, and an aperture diameter of 56 mm. Perfectly matched layers (cylindrical) of 5 mm thickness were applied to the model’s outer boundaries to prevent reflections. The thermal boundary conditions were set as follows: the radiation boundary temperature was fixed at 300 K with thermally insulated boundaries. The transducer was modeled to radiate with a specified vibration velocity. A defined heat source was applied for a duration of 4 s to simulate the heating process. A refined mesh was employed in the near-field of the transducer and within the sample material, with a maximum element size of λ / 30 , where λ is the wavelength of the emitted signal, and the remaining grid size is predefined and hyperfine. The material properties for the polyurethane samples, measured at 25 °C, are listed in Table 1.
Figure 1 presents simulated internal temperature distributions in polyurethane plates with varying Shore hardness (A20–A80) under 13.5W ultrasonic irradiation for 3 s. Samples with lower peak temperatures (e.g., A20 and A50) demonstrate more uniform heating patterns, with elevated temperature regions covering nearly the entire focal zone and penetrating through the material thickness. In contrast, samples exhibiting higher peak temperatures (particularly A70) show concentrated heating primarily at the focal point, displaying distinct conical temperature distributions throughout the material. Analysis of Figure 1 and Table 1 reveals that the Shore A20 polyurethane exhibits the lowest sound velocity and density among all tested samples, corresponding to its minimal temperature rise. While both sound velocity and density generally increase with Shore hardness (except for a notable decrease at A50), the A80 specimen demonstrates significantly higher sound velocity and density yet substantially reduced heating compared to A70, accompanied by distinct morphological characteristics suggesting its glass transition temperature may have surpassed a critical threshold, resulting in fundamentally different thermoacoustic behavior. The maximum temperature elevation observed in the A70 sample likely stems from its optimal combination of molecular chain mobility and enhanced viscous dissipation capacity.
As the acoustic wave propagates through the simulated temperature fields shown in Figure 1, the local variations in sound velocity induce a cumulative phase shift. This phase shift, measured relative to a reference signal unaffected by the thermal gradient, varies depending on the temperature field’s magnitude and distribution. The phase shift magnitude varies with different temperature fields due to temperature-dependent sound velocity variations. Furthermore, ultrasonic wave diffraction occurs when propagating through heterogeneous media or encountering boundary discontinuities, resulting in wavefront distortion and altered phase accumulation. Figure 2a shows the phenomenon of phase shifting due to changes in the speed of sound inside the sample as the sound wave passes through the temperature field. At 42 °C, the phase shift is overall greater than the corresponding value at 35 °C, indicating that the increase in temperature has a significant effect on the phase change in the ultrasonic propagation path, and that the phase shift is greater as the acoustic wave propagates axially within the sample due to phase accumulation.

4. Experiments

4.1. Experimental Setup

The experimental setup is depicted in Figure 3. The excitation system comprised a function generator (RIGOL, DG4062, RIGOL TECHNOLOGIES CO., Ltd., Suzhou, China), a power amplifier (AR, 800A3B, AMETEK, Souderton, PA, USA), and a single-element focused transducer (focal length 9 cm, aperture diameter 5.6 cm, angular aperture 36.2°, operating frequency 600 kHz, Lab-made transducer, Hangzhou, China). The receiving system consisted of a hydrophone probe. Signal analysis was performed using a lock-in amplifier (Sine Scientific Instrument, OE2041, SSI INSTRUMENT, Guangzhou, China) connected to a host computer running custom analysis software (LIA_Console_V1.7.6.231031, SSI INSTRUMENT, Guangzhou, China), which captured the phase and amplitude of the transmitted signal.

4.2. Experimental Procedure

Seven polyurethane plates with Shore hardness ranging from A20 to A80 were used. Prior to testing, their densities were measured at 25 °C. To minimize interference from surface bubbles, all seven standard samples were immersed in degassed water or 30 min. The experimental protocol employed degassed ultrapure water prepared using a dedicated degassing system (DEDU, DMD-50K model) to ensure optimal acoustic transmission properties and eliminate potential interference from dissolved gases.
The experiments employed high-intensity focused ultrasound (HIFU) in the medical frequency range, which induces rapid and significant temperature rise within a short duration. This approach allows for efficient and effective comparison of viscous dissipation index variations across different materials. Each sample was positioned at the focal point of the transducer. To measure viscous loss, a continuous wave (600 kHz, 500 mV) was emitted and amplified to achieve incident acoustic powers of 7.2 W, 13.5 W, and 17 W. The receiving probe was connected to the lock-in amplifier, and the phase and amplitude of the transmitted signal were recorded for 3 s at 0.1 s intervals. The collection point was located approximately 2.5 mm radially from the focal axis. For determining transmission coefficients and sound velocities, a 600 kHz burst signal (50 ms-on, 50 ms-off) was used to analyze time delays and amplitude changes between the received and reference signals.

4.3. Data Inversion and Analysis

The phase variation in a propagating ultrasonic wave is intrinsically linked to the propagation velocity, which is dependent on both material properties and temperature. Consequently, for a given sample subjected to controlled heating at specific power levels, the temperature-dependent phase shift can be measured and utilized to derive the local sound velocity through the following inversion formula. Subsequently, the temperature distribution can be determined based on the established sound velocity-temperature correlation, thereby enabling the conversion of phase differences into temperature gradients. Finally, the viscous loss fraction can be quantitatively calculated from these derived parameters.
Based on the temperature field characteristics of focused ultrasound, according to the basic principle of phase shift temperature measurement of ultrasound transmission signals, it is known that when the emitted ultrasound radially and vertically transmits a non-uniform temperature field from the highest temperature, then the time of ultrasound passing through the non-uniform temperature field can be expressed as:
t L = 2 0 L 1 V T x , t d x
where T ( x , t )   denotes the temperature at the position   x at the time t , V   denotes the sound velocity of ultrasound in the internal temperature field of the sample, and   L is the radius of the radial distribution of the temperature field in the heated part. At this point, the phase shift through the non-uniform temperature field can be expressed as:
Δ φ = 360 × f × 2 L V 0 2 0 L 1 V T x , t d x
where V 0 denotes the sound velocity of sample at room temperature, φ   denotes the phase shift, and f denotes the ultrasonic frequency of the incident probe. When the internal temperature field of the sample changes, the phase shift φ will also change, so the sound velocity at the current temperature can be obtained by detecting the phase shift   φ   and then converted to the current temperature according to the correlation between the temperature and the velocity of sound. The correlation between the temperature and the velocity of sound obtained from experimental measurements is as follows in Figure 4.
Once the temperature rise T is known, the viscous loss Q V L     can be calculated. Based on the simulation results, the isotherm at a radial distance of 2 mm from the focal point is approximately linear along the axial direction. We therefore estimate the viscous loss within this small cylindrical volume. The proportion of viscous loss η V L   relative to the total dissipated energy Q T o t a l is given by:
η V L = Q V L Q T o t a l = c m Δ T E · η F P D
where c is the specific heat capacity of the sample, m is the mass, T   is the temperature difference, η V L   is the percentage of viscous loss, Q V L   is the energy lost to viscous loss, Q T o t a l   is the total energy lost due to scattering and viscous loss, and E is the incident energy.

4.4. Thermocouple Temperature Measurement

To validate the accuracy of the simulated temperature distribution and to confirm that the measured phase shifts are indeed caused by internal temperature variations, we employed thermocouples to monitor the rear surface temperature during ultrasonic irradiation. The experimental measurements were then compared with simulation results. This comparative analysis was conducted to verify that the observed phase shifts were caused by temperature increases, thereby excluding the possibility that these shifts resulted from surface interactions (e.g., acoustic scattering or reflection artifacts) in the high-hardness polyurethane samples.
The validation experiment employed a K-type adhesive thermocouple (KPS-ZT-TT-K-30-2000-LX), measurement range: −40 to 200 °C) attached to the rear surface of the A70 polyurethane plate at the focal region. Temperature measurements were recorded in real-time using a custom LabVIEW program to monitor temperature variations during ultrasonic irradiation. The experimental temperature data were then compared with simulation results. Throughout the experiment, the water temperature was maintained at 27 °C to ensure consistent testing conditions.

5. Results and Discussion

5.1. Phase Shift at Different Temperature

The core of this study involves using phase difference variations in transmitted signals to derive the internal thermal characteristics of the polyurethane plates. Figure 5 shows the real-time phase decrease during 3 s heating intervals at three power levels. As expected, all samples exhibited a phase decrease due to thermal accumulation, with greater phase shifts observed at higher incident powers. The significant variations in phase shift among materials of different Shore hardness highlight the high sensitivity of this method to viscosity differences.

5.2. Inverted and Simulated Temperature Variations

These phase differences were converted to temperature differences, as shown in Figure 6a, and compared with simulated results (Figure 6b) and a direct comparison plot (Figure 6c). The experimental temperature trends closely follow the FPD variations and exhibit an inverse relationship with the transmission coefficients. For a given hardness, temperature differences increase with incident power, confirming a positive correlation between energy input and temperature rise. Higher power levels enhance the sensitivity to hardness variations, yielding more distinct temperature profiles. Notably, dual temperature peaks are observed at A30 and A70 under 7.2 W irradiation, which shift to A40 and A70 at 13.5 W and 17 W. This suggests that the A40 and A70 samples possess superior heat absorption capabilities, with A70 consistently demonstrating the strongest absorption, indicative of a unique viscosity that facilitates material discrimination.
Comparative analysis reveals generally higher simulated temperatures, though trend consistency is maintained. A key discrepancy appears in the low-hardness region at 7.2 W, where experiments show an A30 peak versus simulated A40 peaks across all power levels. This suggests potential low-power measurement interference, implying the existence of a critical minimum power threshold that should be exceeded in future phase-based thermometric measurements. In conclusion, the results collectively demonstrate that this method exhibits sufficient sensitivity to distinguish materials based on their characteristic viscosity.

5.3. Acoustic Performance Metrics and Temperature Validation

As calculated from Equations (10) and (11) and shown in Figure 7, the acoustic parameters exhibit complex, hardness-dependent behavior. The transmission coefficient (Figure 7a) peaks at approximately 0.8 for the A20 sample, decreases, and shows a secondary peak of 0.72 for the A50 sample. Figure 7b an approximately inverse relationship between IL and ER, with ER values being substantially larger than IL. The IL curve displays a bimodal trend with maxima at A40 (5.82 dB) and A70 (7.82 dB), interrupted by a significant drop to 2.77 dB at A50. Conversely, ER peaks at A20 (42.21 dB) and follows a complex pattern. The anomalous behavior observed for the A50 sample in both transmission and loss parameters suggests it possesses distinctive acoustic-mechanical coupling properties compared to the other specimens.
The FPD, calculated using Equation (12), is shown in Figure 8. Its trend closely mirrors that of the insertion loss, confirming that transmitted energy loss is the dominant dissipation pathway compared to reflected energy. The FPD increases from A20 to a sub-peak at A40 (73%), drops to a minimum at A50 (47%), and then rises to a maximum at A70 (83%) before declining for A80. This bimodal distribution suggests complex, hardness-dependent energy dissipation mechanisms.
To verify that the phase shifts were indeed caused by temperature increases and to exclude the possibility that these shifts resulted from surface interactions in the high-hardness polyurethane samples, the measured temperature data during ultrasonic heating were compared with simulation results as shown in Figure 9 below.
As shown in Figure 9, the temperature measured at the material’s rear surface exhibits a gradual increase during ultrasonic irradiation, with higher input power leading to greater temperature elevation. The experimental temperature profiles show good agreement with numerical simulations, demonstrating consistency between the simulated internal temperature field and actual thermal distribution within the material. This correlation further confirms that the observed phase variations originate from viscous effects rather than surface interactions.

5.4. Polymer Viscious Dissipation Index

The proportion of total power dissipation attributable to viscous heating is presented in Figure 10 for both experimental and simulation results. Both plots confirm that viscous dissipation is the predominant energy loss mechanism, with trends that are similar to the FPD and IL curves. The experimental data at 13.5 W and 17 W show comparable patterns: a gradual decline from A20 to A60 (with an exception at A40) followed by a sharp increase to a maximum viscous loss at A70. The A70 sample reached a maximum viscous loss proportion of approximately 92.4% at 13.5 W. The 7.2 W results show significant deviations, likely due to measurement noise at low power, and are considered less reliable. In contrast, the simulated viscous loss ratios consistently exceeded 86%, showing good agreement with experimental trends and maintaining characteristic peaks at A40 and A70. Notably, the simulated A70 specimen reached an exceptional viscous loss proportion of 97.6% under 17 W acoustic excitation.
The results demonstrate strong consistency between the experimental measurements and simulations, confirming the validity of this ultrasonic method as a novel approach for distinguishing polymer viscous dissipation index.

6. Conclusions

Current methods for measuring polymer viscosity are predominantly limited to low-frequency conditions, leaving a gap in effective high-frequency characterization. This study presents an improved methodology that accurately characterizes the polymer viscous dissipation index under high-frequency ultrasound. By employing phase-shift thermometry, we convert measured phase variations in transmitted signals—resulting from minute sound velocity changes during heating—into temperature profiles. Subsequent thermodynamic analysis determines the proportion of viscous loss in the total energy dissipation, providing an effective metric for the material viscous dissipation index. Due to the inherent limitations of the underlying principle, the proposed method is only applicable to viscoelastic materials that exhibit temperature-dependent sound velocity variations and possess homogeneous internal structures without cavity or bubble interference. However, this method is unsuitable for high-viscosity viscoelastic materials (e.g., η   > 10 4   P a · s ), as ultrasonic heating may merely induce localized softening without producing significant sound velocity changes. Furthermore, the method cannot be applied to non-Newtonian materials such as toothpaste or blood.
Experimental and simulation results demonstrate that the phase-shift thermometry method achieves high measurement accuracy, revealing a consistent temperature–hardness correlation. However, it is essential to properly control both the minimum incident power to reduce experimental interference and the applied ultrasound frequency. Excessive frequency levels would increase the Deborah number of polymer chains, resulting in viscosity reduction that significantly compromises measurement accuracy. While the calculated viscous loss values show some deviation from simulated results, the overall trends are well-matched.
This study successfully used Shore hardness to differentiate polyurethanes, recognizing its correlation with intrinsic material characteristics like viscosity. The method can be extended to evaluate materials with well-defined mechanical properties [28,29] (e.g., tensile modulus, elongation at break), chemical compositions (e.g., polyether vs. polyester base), molecular weights, and crosslink densities, enabling a more comprehensive, viscosity-based material assessment. The established correlations between viscous loss and acoustic parameters like transmission coefficient and insertion loss further strengthen its potential as a robust characterization tool.

Author Contributions

Conceptualization, Y.W. and Z.Y.; methodology, Z.L. and Z.Y.; software, Y.W.; validation, Z.Y.; formal analysis, Z.Y.; investigation, Z.Y.; resources, Y.W.; data curation, Y.W. and Z.Y.; writing—original draft preparation, Z.Y.; writing—review and editing, Z.Y. and Y.W.; visualization, Z.Y.; supervision, Y.W.; project administration, Y.W.; funding acquisition, Y.W. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Key Research and Development Program of China, grant number (2022YFF0607505) and the APC was funded by China Jiliang University.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Acknowledgments

The authors gratefully acknowledge the assistance provided by the staff of the Key Laboratory of in situ Metrology for Education Ministry at China Jiliang University in the preparation of this work.

Conflicts of Interest

The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Abbreviations

The following abbreviations are used in this manuscript:
FPDFractional-Power-Dissipation

References

  1. Rensselar, V.J. Ultrasound monitoring. Tribol. Lubr. Technol. 2021, 77, 28–31. [Google Scholar]
  2. Ono, K.A. Comprehensive Report on Ultrasonic Attenuation of Engineering Materials, Including Metals, Ceramics, Polymers, Fiber-Reinforced Composites, Wood, and Rocks. Appl. Sci. 2020, 10, 2230. [Google Scholar] [CrossRef]
  3. Wang, D.; Cai, S.; Wang, H.; Chen, Z.; Huang, H.; Chen, D.; Wu, C. Study on ultrasonic lamb wave testing technology of honeycomb sandwich structures. J. Phys. Conf. Ser. 2024, 2822, 012140. [Google Scholar] [CrossRef]
  4. Mascarenhas, A.R.P.; de Melo, R.R.; Pimenta, A.S.; Stangerlin, D.M.; de Oliveira Corrêa, F.L.; Sccoti, M.S.V.; de Oliveira Paula, E.A. Ultrasound to estimate the physical-mechanical properties of tropical wood species grown in an agroforestry system. Holzforschung 2021, 75, 879–891. [Google Scholar] [CrossRef]
  5. Zhu, Y.; Dong, C.; Yin, Y.; Chen, X.; Guo, Y.; Zheng, Y.; Shen, Y.; Wang, T.; Zhang, X.; Chen, S. The role of viscosity estimation for oil-in-gelatin phantom in shear wave based ultrasound elastography. Ultrasound Med. Biol. 2015, 41, 601–609. [Google Scholar] [CrossRef]
  6. Irfan, M.; Farooq, M.A. Thermophoretic MHD free stream flow with variable internal heat generation/absorption and variable liquid characteristics in a permeable medium over a radiative exponentially stretching sheet. J. Mater. Res. Technol. 2020, 9, 4855–4866. [Google Scholar] [CrossRef]
  7. Maturana, L.G.; López, A.O. Determination and assessment of the linear viscoelastic range and viscoelastic properties of modified asphalt and mastics under different temperature conditions. Constr. Build. Mater. 2024, 420, 135606. [Google Scholar] [CrossRef]
  8. Genovés, V.; Maini, L.; Roman, C.; Hierold, C.; Cesarovic, N. Variation in the viscoelastic properties of polydimethylsiloxane (PDMS) with the temperature at ultrasonic frequencies. Polym. Test. 2023, 124, 108067. [Google Scholar] [CrossRef]
  9. Formigoni, P.O.; Franco, E.E.; Reyna, C.A.; Tsuzuki, M.S.; Buiochi, F. Viscoelastic characterization of liquids using ultrasonic shear-waves generated by the internal reflection approach. Ultrasonics 2025, 153, 107643. [Google Scholar] [CrossRef]
  10. Padmanaban, R.; Gayathri, A.; Gopalan, A.I.; Lee, D.-E.; Venkatramanan, K. Comparative Evaluation of Viscosity, Density and Ultrasonic Velocity Using Deviation Modelling for Ethyl-Alcohol Based Binary Mixtures. Appl. Sci. 2023, 13, 7475. [Google Scholar] [CrossRef]
  11. Mason, W.P.; Baker, W.O.; Mcskimin, H.J.; Heiss, J.H. Measurement of shear elasticity and viscosity of liquids at ultrasonic frequencies. Phys. Rev. 1949, 75, 936–946. [Google Scholar] [CrossRef]
  12. Balasubramaniam, K.; Shah, V.V.; Costley, R.D.; Boudreaux, G.; Singh, J.P. High temperature ultrasonic sensor for the simultaneous measurement of viscosity and temperature of melts. Rev. Sci. Instrum. 1999, 70, 4618–4623. [Google Scholar] [CrossRef]
  13. Waqas, H.; Yasmin, S.; Muhammad, T.; Imran, M. Flow and heat transfer of nanofluid over a permeable cylinder with nonlinear thermal radiation. J. Mater. Res. Technol. 2021, 14, 2579–2585. [Google Scholar] [CrossRef]
  14. Teixeira, M.C.C.; Pereira, M.V.S.; Souza, R.F.M.; Lopes, F.R.; da Silva, T.G. Assessment of Fatigue Life and Failure Criteria in Ultrasonic Testing Through Thermal Analyses. Appl. Sci. 2025, 15, 1076. [Google Scholar] [CrossRef]
  15. Rabbani, A.; Schmitt, D.R. Ultrasonic shear wave reflectometry applied to the determination of the shear moduli and viscosity of a viscoelastic bitumen. Fuel 2018, 232, 506–518. Available online: https://www.sciencedirect.com/science/article/pii/S0016236118310202 (accessed on 6 May 2025). [CrossRef]
  16. Shuo, L.; Lvping, R.E.N.; Chunlan, Y. Study of thermometry based on time shift of ultrasound echo. J. Press China Med. Devices 2009, 24, 12–14. [Google Scholar]
  17. Miller, N.R.; Bamber, J.C.; Haar, G.R.T. Imaging of temperature-induced echo strain: Preliminary in vitro study to assess feasibility for guiding focused ultrasound surgery. Ultrasound Med. Biol. 2004, 30, 345–356. [Google Scholar] [CrossRef] [PubMed]
  18. Gertner, M.R.; Worthington, A.E.; Wilson, B.C.; Sherar, M.D. Ultrasound imaging of thermal therapy in in vitro liver. Ultrasound Med. Biol. 1998, 24, 1023–1032. [Google Scholar] [CrossRef] [PubMed]
  19. He, J.; Lee, S.S.; Kalyon, D.M. Shear viscosity and wall slip behavior of dense suspensions of polydisperse particles. J. Rheol. 2019, 63, 19–32. [Google Scholar] [CrossRef]
  20. Gholami, M.; Khodadadi, M.; Hajikarimi, P.; Khodaii, A. Investigating the effects of reducing the number of temperatures and frequencies on the development of master curves for viscoelastic properties of bituminous composite. Measurement 2024, 230, 114503. [Google Scholar] [CrossRef]
  21. Ferry, J.D. Viscoelastic Properties of Polymers, 3rd ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 1961; pp. 1–96. [Google Scholar]
  22. Brinson, H.F.; Brinson, L.C. Polymer Engineering Science and Viscoelasticit: An Introduction; Springer: Boston, MA, USA, 2008; pp. 3–45. [Google Scholar]
  23. Jiang, B.; Peng, H.; Wu, W.; Jia, Y.; Zhang, Y. Numerical Simulation and Experimental Investigation of the Viscoelastic Heating Mechanism in Ultrasonic Plasticizing of Amorphous Polymers for Micro Injection Molding. Polymers 2016, 8, 199. [Google Scholar] [CrossRef]
  24. Tsui, P.H.; Chien, Y.T.; Liu, H.L.; Shu, Y.C.; Chen, W.S. Using ultrasound CBE imaging without echo shift compensation for temperature estimation. Ultrasonics 2012, 52, 925–935. [Google Scholar] [CrossRef]
  25. Pennes, M. Analysis of tissue and arterial blood temperature in the resting human foearm. J. Appl. Physiol. 1948, 1, 93–122. [Google Scholar] [CrossRef]
  26. Kinsler, L.E.; Frey, A.R.; Coppens, A.B.; Sanders, J.V. Fundamentals of Acoustics, 4th ed.; Wiley: Hoboken, NJ, USA, 2000; pp. 1–45. [Google Scholar]
  27. Precision Acoustics Ltd. Apt Flex F48 Technical Data Sheet; Acoustic Polymers Ltd.: Dorchester, UK, 2023; pp. 1–5. Available online: https://www.acoustics.co.uk (accessed on 26 April 2025).
  28. Somdee, P.; Lassú-Kuknyó, T.; Kónya, C.; Szabó, T.; Marossy, K. Thermal analysis of polyurethane elastomers matrix with different chain extender contents for thermal conductive application. J. Therm. Anal. Calorim. 2019, 138, 1003–1010. [Google Scholar] [CrossRef]
  29. Shen, M.; Huang, Q. Acoustic velocity and attenuation coefficient of magnetorheological fluids under electromagnetic fields. Appl. Acoust. 2016, 107, 27–33. [Google Scholar] [CrossRef]
Figure 1. Simulated internal temperature distributions in polyurethane plates of varying Shore hardness after 3 s of 13.5 W ultrasonic irradiation. (a) Shore A20; (b) Shore A30; (c) Shore A40; (d) Shore A50; (e) Shore A60; (f) Shore A70; (g) Shore A80. The images reveal that samples with higher viscous loss (e.g., A70) exhibit more localized, conical heating patterns.
Figure 1. Simulated internal temperature distributions in polyurethane plates of varying Shore hardness after 3 s of 13.5 W ultrasonic irradiation. (a) Shore A20; (b) Shore A30; (c) Shore A40; (d) Shore A50; (e) Shore A60; (f) Shore A70; (g) Shore A80. The images reveal that samples with higher viscous loss (e.g., A70) exhibit more localized, conical heating patterns.
Applsci 15 09267 g001
Figure 2. (a) Comparison of received acoustic signals with and without passing through a heated temperature field. (b) Comparison of the cumulative phase shift for signals passing through fields at different temperatures (35 °C and 42 °C).
Figure 2. (a) Comparison of received acoustic signals with and without passing through a heated temperature field. (b) Comparison of the cumulative phase shift for signals passing through fields at different temperatures (35 °C and 42 °C).
Applsci 15 09267 g002
Figure 3. (a) Photograph and (b) schematic diagram of the experimental setup for ultrasonic viscous dissipation index measurement.
Figure 3. (a) Photograph and (b) schematic diagram of the experimental setup for ultrasonic viscous dissipation index measurement.
Applsci 15 09267 g003
Figure 4. Experimentally measured relationship between sound velocity and temperature for the seven polyurethane samples.
Figure 4. Experimentally measured relationship between sound velocity and temperature for the seven polyurethane samples.
Applsci 15 09267 g004
Figure 5. Phase change measured in real time by heating for 3 s. (a) Incident acoustic power 7.2 W; (b) Incident acoustic power 13.5 W; (c) Incident acoustic power 17 W.
Figure 5. Phase change measured in real time by heating for 3 s. (a) Incident acoustic power 7.2 W; (b) Incident acoustic power 13.5 W; (c) Incident acoustic power 17 W.
Applsci 15 09267 g005
Figure 6. Temperature rise trends as a function of Shore hardness. (a) Experimental temperature rise. (b) Simulated temperature rise. (c) Direct comparison of experimental (solid lines) and simulated (dotted lines) results.
Figure 6. Temperature rise trends as a function of Shore hardness. (a) Experimental temperature rise. (b) Simulated temperature rise. (c) Direct comparison of experimental (solid lines) and simulated (dotted lines) results.
Applsci 15 09267 g006
Figure 7. Measured acoustic parameters as functions of Shore hardness. (a) Transmission coefficient versus hardness. (b) Comparison of Insertion Loss (IL) and Echo Reduction (ER) versus hardness.
Figure 7. Measured acoustic parameters as functions of Shore hardness. (a) Transmission coefficient versus hardness. (b) Comparison of Insertion Loss (IL) and Echo Reduction (ER) versus hardness.
Applsci 15 09267 g007
Figure 8. Fractional-Power-Dissipation results for the seven polyurethane samples.
Figure 8. Fractional-Power-Dissipation results for the seven polyurethane samples.
Applsci 15 09267 g008
Figure 9. Actual temperature change (solid line) and simulated temperature change (dotted line).
Figure 9. Actual temperature change (solid line) and simulated temperature change (dotted line).
Applsci 15 09267 g009
Figure 10. Proportion of total power dissipation attributable to viscous heating. (a) Experimental results; (b) Simulation results.
Figure 10. Proportion of total power dissipation attributable to viscous heating. (a) Experimental results; (b) Simulation results.
Applsci 15 09267 g010
Table 1. Material parameters for this simulation.
Table 1. Material parameters for this simulation.
HardnessSound Velocity (m/s)Density (kg/m3)Attenuation Coefficient (Np/m)
A201476106110
A301538108925
A401560109431
A501542107813
A601565109832
A701664111644
A801869126426
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Yang, Z.; Wang, Y.; Lu, Z. Research on Viscous Dissipation Index Assessment of Polymer Materials Using High-Frequency Focused Ultrasound. Appl. Sci. 2025, 15, 9267. https://doi.org/10.3390/app15179267

AMA Style

Yang Z, Wang Y, Lu Z. Research on Viscous Dissipation Index Assessment of Polymer Materials Using High-Frequency Focused Ultrasound. Applied Sciences. 2025; 15(17):9267. https://doi.org/10.3390/app15179267

Chicago/Turabian Style

Yang, Zeqiu, Yuebing Wang, and Zhenwei Lu. 2025. "Research on Viscous Dissipation Index Assessment of Polymer Materials Using High-Frequency Focused Ultrasound" Applied Sciences 15, no. 17: 9267. https://doi.org/10.3390/app15179267

APA Style

Yang, Z., Wang, Y., & Lu, Z. (2025). Research on Viscous Dissipation Index Assessment of Polymer Materials Using High-Frequency Focused Ultrasound. Applied Sciences, 15(17), 9267. https://doi.org/10.3390/app15179267

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop