Radially Poled Tubular Piezoceramic

• Radially poled tubular piezoceramics are piezoelectric tubes with radially aligned domains that efficiently excite fundamental vibration modes to generate focused acoustic fields.
• The design utilizes functionally graded materials and analytical modeling to precisely control stress, strain, and resonance properties under high power inputs.
• These transducers enhance sonoprocessing and sonochemistry by providing uniform cavitation distribution and superior volumetric power density compared to conventional sonotrodes.

A radially poled tubular piezoceramic is a piezoelectric tube—typically composed of ferroelectric ceramic—subjected to an electric poling process that aligns its ferroelectric domains in the radial direction. This configuration enables efficient excitation of fundamental radial vibration modes, resulting in inwardly focusing acoustic fields. Such transducers are integral to modern high-intensity ultrasound applications, particularly in flow-based sonochemistry and sonoprocessing systems, where they offer superior cavitation distribution compared to conventional Langevin sonotrodes. Analytical modeling, both in homogeneous and functionally graded contexts, enables precise control of stress, strain, and electric field distributions, meeting the demands of power ultrasonics and advanced actuator design (Li et al., 5 Dec 2025, Wang et al., 2018).

1. Materials, Construction, and Electrode Configuration

Radially poled tubular piezoceramics are typically fabricated from lead-zirconate-titanate (PZT) alloys, such as Ferroperm PZ26, characterized by relative permittivity $\varepsilon_r \simeq 1\,800$, density $\rho \simeq 7\,500\,\mathrm{kg/m^3}$, and Young’s modulus $E \simeq 70\,\mathrm{GPa}$. Standard dimensions, as reported in "The tube transducer as a novel source for power ultrasound" (Li et al., 5 Dec 2025), include:

Parameter	Symbol	Typical Value
Outer diameter	$D_o$	$63.4\,\mathrm{mm}$
Inner diameter	$D_i$	$55.6\,\mathrm{mm}$
Wall thickness	$t$	$3.9\,\mathrm{mm}$
Axial length	$L$	$30.3\,\mathrm{mm}$

Full-circumference silver electrodes are coated on the inner and outer curved surfaces, but not on the tube ends. Electrical connections are made via soldered wires to the coatings, subsequently insulated to prevent shorting.

For functionally graded generations, tubular composites such as PZT4/PVDF (polyvinylidene fluoride) are synthesized with spatially varying properties, as detailed in (Wang et al., 2018). The spatial variation of composition and thus piezoelectric and elastic constants is governed by graded power-law functions.

2. Radial Poling Process and Field Alignment

Poling in the radial direction is achieved by applying a high DC voltage ($V_p \approx 2$–$4\,\mathrm{kV}$) across the wall thickness at elevated temperature ($T_{pole} \approx 100$–$120\,^\circ$C) for a duration of $10$–$30\,\min$. The inner and outer electrodes are used as terminals, and the ceramic is unconstrained at its ends. The resulting electric field

$E_r(r) = -\frac{dV_p}{dr}$

is approximately uniform ($E_r \simeq V_p/t$) in thin-walled geometries. This field orients the ferroelectric domains radially, establishing a primary piezoelectric coefficient $d_{31,r}$ that generates circumferential and axial strain under subsequent excitation.

In functionally graded cylinders, the spatial variation of properties entails a non-uniform field and more complex electric and mechanical response, systematically derivable from the graded constitutive relations (Wang et al., 2018).

3. Vibrational Modes and Resonance Properties

The electromechanical behavior of radially poled tubes is dominated by fundamental radial ("breathing") and axial modes.

Radial Resonance Mode

The governing equation for radial displacement $u(r)$ is: $$\frac{1}{r}\frac{d}{dr}\left[r\frac{d u}{dr}\right] + \frac{\rho}{E}\omega^2 u = 0$$ with general solution

$$u(r) = A J_1(k r) + B Y_1(k r),\quad k^2 = \frac{\rho}{E}\omega^2$$

Boundary conditions (traction-free at $r=R_i, R_o$) yield the resonance condition: $J_0(k R_o)Y_0(k R_i) - J_0(k R_i)Y_0(k R_o) = 0$ For thin walls, the fundamental frequency approximates: $$f_r \simeq \frac{\chi_{01}}{2\pi R_{avg}} \sqrt{\frac{E}{\rho}}$$ where $\chi_{01} \approx 2.405$, $R_{avg} = (R_o + R_i)/2$.

Experimentally, (Li et al., 5 Dec 2025) recorded $f_{r1} = 16.79\,\mathrm{kHz}$.

Axial Resonance Mode

For free tube ends, the lowest axial resonance is given by: $$f_{z1} = \frac{c_L}{2L},\quad c_L = \sqrt{\frac{E}{\rho}}$$ with $f_{z1}$ measured at $55.89\,\mathrm{kHz}$ in (Li et al., 5 Dec 2025).

Functionally Graded Dynamics

For radially poled, functionally graded tubes, dynamic and static response is obtained from hypergeometric solutions to the graded elasticity–piezoelectricity system: $$x^2(x-1)\frac{d^2u}{dx^2} + x\left[2x-1+\left(2-\frac{2}{n}\right)\right]\frac{du}{dx} - \left( \frac{\Phi_3}{\Phi_1} x - \frac{\Phi_2}{\Phi_1} \right) u = -\frac{C_6}{n^2\Phi_1}x^{ -\frac{2}{n}-1 }$$ where coefficients $\Phi_{1,2,3}$ depend on graded material constants and the profile exponent $n$ (Wang et al., 2018).

4. Electromechanical Circuit Modeling

The behavior near specific vibrational modes can be captured by the Butterworth–Van Dyke (BvD) equivalent circuit model. The parallel static capacitance,

$$C_0 = \varepsilon_0 \varepsilon_r \frac{A_{eff}}{t}$$

(with $A_{eff} \simeq 2\pi R_{avg} L$), is in parallel with a motional branch comprising $R_m$, $L_m$, and $C_m$. The model provides resonance ($\omega_r$) and antiresonance ($\omega_p$) features, with coupling factor

$k_r^2 = \frac{\omega_p^2 - \omega_s^2}{\omega_p^2}$

and mechanical quality factor

$Q_m = \omega_r L_m / R_m$

This model enables efficient prediction and engineering of electrical input–acoustic output transfer characteristics (Li et al., 5 Dec 2025).

5. Acoustic Field Topology and Cavitation Phenomena

Excitation of the radial mode results in converging radial displacement and associated pressure fields, focusing maximum acoustic pressure on the tube axis. The time-varying pressure: $p(r,t) \simeq p_0\, J_0(k_r r)\cos(\omega t)$ produces volume-filling, axially centered cavitation. Experimental quantification via sonochemiluminescence (SCL) and high-speed imaging demonstrated that, at 55 W input, $\sim97\%$ of the tube bore achieves high-intensity SCL levels, exceeding $99\%$ at 106 W. Maximum image intensity reached up to 190 compared to 120 for the sonotrode tip (Li et al., 5 Dec 2025). High-speed video reveals persistent, dense filamentary bubble structures and dominant on-axis clusters, indicating intense cavitation conditions. The transducer can exceed cavitation thresholds ($P_{th} \simeq 1$–$2$ bar in water at 20 kHz) throughout the entire bore.

6. Comparison with Conventional Sonotrodes and Advantages

A direct comparison with 20 kHz Langevin-type sonotrodes shows the tubular transducer achieves greater volumetric power density (0.75 W/mL versus 0.28 W/mL at 55 W for tube and sonotrode, respectively) and distributes cavitation more uniformly. Sonotrode activity is highly localized beneath the tip, while the tubal design maintains near-uniform intensity throughout its bore, with no concentration of erosive activity on the ceramic wall itself (Li et al., 5 Dec 2025). Mechanical $Q$-factors are comparable ($Q \approx 35$ for the tube; $Q \approx 20$–$50$ for the sonotrode). At similar input powers, the tube transducer delivers earlier cavitation onset and greater peak intensities due to its focused pressure fields.

7. Applications and Design in Sonoprocessing and Ultrasonics

Radially poled tubular piezoceramics are well suited to modular flow-through reactors for continuous, high-throughput sonoprocessing. Arrays of tubes can be inserted coaxially into industrial piping, each acting as a "cavitation module." Design considerations include hydraulic losses (favoring smooth, flush-mounted tubes), avoidance of acoustic interference (appropriate tube spacing and drive phase control), and ensuring full flow exposure to on-axis high-intensity regions. Material processed within the bore is exposed to distributed high-intensity cavitation while leaving the ceramic walls protected from direct bubble activity and associated erosion. The use of functionally graded tube structures allows further optimization of stress distribution and resonance properties, with active control over spatial compliance and mitigation of stress concentrations (Wang et al., 2018).

The integration of analytical solutions for graded structures enables tuning of static and dynamic strain fields, allowing resonance placement and performance tailored to specific process requirements. Cumulatively, radially poled tubular piezoceramic transducers are central to the development of scalable power ultrasonic systems for material processing, separation, and advanced actuation (Li et al., 5 Dec 2025, Wang et al., 2018).