Cavitation-eroded piezoelectric ceramics as finite mechano-electrochemical transducers

Graphical abstract

Abstract

Piezoelectric ceramics are being explored as mechanically powered materials for energy-environmental processes, but the relationship between energy transduction and material durability remains unclear. We examined four piezoelectric materials (BaTiO₃, ZnO, K_0.5Na_0.5NbO₃, and PbZr_0.53Ti_0.47O₃) under 40 kHz ultrasonic cavitation. The pressure criterion for reactive oxygen species (ROS) formation through water compression (2.8–8.1 GPa) exceeded the 1.13 GPa compressed-bubble pressure obtained from the bubble-collapse calculation. The electron-energy analysis provides a thermal-equilibrium reference rather than a prediction of plasma-derived ROS yields. The powder-free control showed only 0.7% and 1.9% tetracycline (TC) removal after 120 and 480 min, respectively. The open-circuit piezoelectric potentials calculated for 1 μm particles at the 1.13 GPa compressed-bubble pressure (9–210 V) exceeded the electrochemical thresholds (0.33–2.8 V), and the powders showed persistent contact-potential shifts up to − 318 mV and charge storage 32–55 times beyond the electrical double-layer limit. Cavitation-induced irreversible lattice strain and selective A-site ion leaching were accompanied by elbow-shaped TC degradation kinetics with performance declining from approximately 67% to 1–3% over five cycles. Mott–Schottky and impedance analyses indicated that interfacial electrochemical states were irreversibly altered. The tested powders are described here as finite mechano-electrochemical transducers whose operation is accompanied by irreversible material changes and declining activity. The relevant design criterion for sustainable piezoelectric environmental applications should include redox output, material lifetime, recoverability, and ion-release risk as well as the first-cycle activity.

1. Introduction

Ultrasonication (US), mechanical vibration, and fluid flow act as driving sources that activate piezoelectric materials in water [1], [2], [3], [4]. Piezoelectric materials with non-centrosymmetric crystal structures generate dipole moments and internal electric fields under mechanical stimuli, such as pressure, vibration, and so on [5]. The internal electric fields generated in aqueous media produce reactive oxygen species (ROS) that promote organic contaminant degradation in water [6], [7], [8], [9]. BaTiO₃ produced H₂ at 1250 μmol·h⁻¹ and 95% of 10 mg·L⁻¹ methylene blue was degraded within 3 h under 40 kHz and 100 W US [10]. Mechanical vibration of 0.15 g BaTiO₃ ceramic powders at 200 rpm in a ten Zr ball mill was used to achieve about 61% degradation of 5 mg·L⁻¹ methylene blue [11]. Fluid-flow to 2D MoS₂ in a helical tube reactor caused 94.8% degradation of 10 mg·L⁻¹ benzothiazole within 2.4 min [12]. These were reported as piezocatalysis because ROS generation and organic contaminant degradation resembled photocatalytic advanced oxidation processes (AOPs) [13], [14], [15], [16].

US is a representative mechanical energy source that induces acoustic cavitation and locally amplifies physical loads in aqueous reaction systems [17], [18]. Ultrasound passing through water creates continuous cycles of low pressure (rarefaction) and high pressure (compression) [19]. Bubbles grow, form, and collapse when the acoustic pressure amplitude reaches the cavitation threshold, which exceeds the tensile strength of the liquid and the stability of microbubble nuclei in the medium [20]. Low-frequency ranges (tens of kHz) exceed the Blake threshold at relatively low energy inputs, so microbubble nuclei grow explosively and form larger cavities [21]. Larger bubble collapse generates higher mechanical shock and shear, maximizing mechanical stimuli applied to material surfaces. The acoustic wave delivers little of this load by itself. Ultrasonic vibration alone generates less than 0.02 V on a piezoelectric particle, whereas the shockwave from a collapsing bubble generates 6.6 V [4]. Therefore, the material damages are from the cavitation intensity [22]. Mechanical load on the solid and chemical yield of the liquid move in opposite directions with frequency. Bubble collapse at tens of kHz supplies the stress that polarizes the ceramic and the stress that damages it.

Two theories can be applied to explain how the polarization drives the interfacial redox [23]. The energy band theory was inspired from photocatalysis [24]. It attributes the ability of piezocatalysts to their band-edge positions and states that piezoelectric potentials distort band structures and drive electrons/holes toward the surface. The energy band theory assumes that piezoelectric materials are semiconductors with bandgaps lower than 4.0 eV and the bandgap width can be decreased under stress [25]. However, this theory cannot explain how polymer piezoelectric materials, such as polyvinylidene fluoride (PVDF), with an insulator-level bandgap of ∼ 5.99 eV, generate ROS in water. It also fails to explain how electrons and holes that were not separated by excitation flow down to the surface along a piezoelectrically-bent bandgap in semiconductors with bandgaps below 4.0 eV [26]. The screening charge effect theory offers different explanation [27]. This framework states that internal electric fields generated by piezoelectricity provide direct thermodynamic energy at the interface, which converts interfacial ions and water molecules into ROS. The main difference between the two theories lies in where the redox reactions originate from: inside the material or outside the material surface. The screening charge effect describes the conversion of mechanical work into an electrical driving force for interfacial redox reactions. Both frameworks focus on how polarization drives these reactions, but neither establishes how the solid retains its structural and electrochemical function during repeated cavitation.

This research examines whether ceramic piezoelectric materials under cavitation from 40 kHz US in water function as finite mechano-electrochemical transducers whose operation is accompanied by irreversible material changes. This question was examined using representative ceramic piezoelectric materials: BaTiO₃, ZnO, potassium sodium niobate (K_0.5Na_0.5NbO₃, KNN), and lead zirconate titanate (PbZr_0.53Ti_0.47O₃, PZT). Pressure-dependent thermodynamics, surface electrochemical analysis, and tetracycline (TC) degradation results were used to evaluate mechano-electrochemical energy conversion and the accompanying material changes.

2. Results and Discussion

2.1. Thermodynamic pathway for ROS generation

Standard Gibbs free energies (ΔG°) are calculated as:

$$\mathrm{\Delta }{\mathrm{G}}^{°}=-\mathrm{n}·\mathrm{F}·{\mathrm{E}}^{°}(\mathrm{k}\mathrm{J}·\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1})$$ (1)

where n is the number of moles of electrons transferred in the redox reaction, F is the Faraday constant (96,485C·mol⁻¹), and E° is the standard potential. Standard potentials for representative ROS generation reactions in water–oxygen systems are E°(O₂/·O₂^–) =  − 0.33 V, E°(·O₂^–/¹O₂) = +0.8 V, and E°(H₂O/·OH) = +2.8 V vs. normal hydrogen electrode (NHE), respectively. The values of ΔG° from Eqn. (1) are 31.8, 77.1, and 270.2 kJ·mol⁻¹, respectively, indicating endothermic reactions [28], [29], [30]. These energies should be supplied to piezomaterials for each ROS generation. The mechanical energy by water pressure (W_mech) is:

$${\mathrm{W}}_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}\approx \frac{\mathrm{\Delta }\mathrm{P}}{2}·\left(\mathrm{\beta }·{\mathrm{V}}_{\mathrm{m}}·\mathrm{\Delta }\mathrm{P}\right)=\frac{1}{2}\mathrm{\beta }·{\mathrm{V}}_{\mathrm{m}}·\mathrm{\Delta }{\mathrm{P}}^2(\mathrm{k}\mathrm{J}·\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1})$$ (2)

where ΔP is the applied pressure, β is the isothermal compressibility of water at standard temperature and pressure (4.6E-10 Pa⁻¹), and V_m is the molar volume of water (1.8E-05 m³∙mol⁻¹) [31]. Solving Eqn. (2) for ΔP gives the pressures required to supply these energies, which are 2.8, 4.3, and 8.1 GPa for E°(O₂/·O₂^–) =  − 0.33 V, E°(·O₂^–/¹O₂) = +0.8 V, and E°(H₂O/·OH) = +2.8 V vs. NHE, respectively. Therefore, ROS starting from O₂/·O₂^– could be generated by water pressure alone if 40 kHz US cavitation produces pressures exceeding 2.8 GPa. Piezoelectric materials would then function as reaction rate enhancers.

The initial bubble diameter of 5.56 μm was measured immediately after 40 kHz US. The Keller–Miksis equation, which models nonlinear bubble dynamics with liquid compressibility corrections, was used with this diameter to simulate a single bubble collapse [32]. The compressed-bubble state used for the subsequent analyses had a diameter of 0.66 μm, an internal pressure of 1.13 GPa, and a gas temperature of 3,751 K (Fig. 1(a, b) and S1). The internal pressure of this state was below the 2.8 GPa pressure criterion calculated using Eqn. (2). The stress-transfer calculation was initialized with this compressed-bubble state, and the thermal-equilibrium electron-energy estimate used its gas temperature.

Fig. 1.

(a) Pressure and (b) Temperature distributions around the compressed bubble, (c) Equilibrium Maxwell–Boltzmann electron energy distribution at 3,751 K, and (d) Critical stresses for ROS generation by BaTiO₃, ZnO, KNN, and PZT.

Cavitation can generate transient plasma [33]. The Maxwell–Boltzmann distribution at 3,751 K was used as a thermal-equilibrium reference for the electron-energy analysis:

$$\mathrm{f}\left(\mathrm{E}\right)=\frac{2}{\sqrt{\mathrm{\pi }}}·\frac{1}{{({\mathrm{k}}_{\mathrm{B}}\mathrm{T})}^{3/2}}·\sqrt{\mathrm{E}}·\exp \left(-\frac{\mathrm{E}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)$$ (3)

$$\mathrm{P}(\mathrm{E}\ge {\mathrm{E}}^{°})={\int }_{{\mathrm{E}}^{°}}^{\infty }\mathrm{f}(\mathrm{E})\mathrm{d}\mathrm{E}$$ (4)

where k_B is the Boltzmann constant (1.381E-23 J·K⁻¹), T is the specified gas temperature of 3,751 K, and P(E ≥ E°) is the fraction of electrons above the corresponding energy criterion E°. The equilibrium fractions are 0.564, 0.176, and 6.060E-04 for O₂/·O₂^–, ·O₂^–/¹O₂, and H₂O/·OH, respectively (Fig. 1(c)). These values are reference electron fractions rather than probabilities of ROS formation. The pressure-only route did not meet the energy criteria in Eqn. (2) at the compressed-bubble state. A plasma-mediated route can contribute to ROS formation [34], [35], but its contribution cannot be inferred from these equilibrium fractions.

Another possible mechanism is the piezoelectric pathway, where cavitation-induced stress generates an electric potential at the solid–water interface instead of compressing water directly. Piezoresponse force microscopy (PFM) results of BaTiO₃, ZnO, KNN, and PZT indicated piezoelectric coefficients (d₃₃) of 124, 14, 111, and 210 pC·N⁻¹, respectively (Eqn. (9) and Fig. S2). Piezoelectric voltage constants (g₃₃) were calculated as:

$${\mathrm{g}}_{33}=\frac{{\mathrm{d}}_{33}}{{\mathrm{\epsilon }}_0·{\mathrm{\epsilon }}_{\mathrm{r}}}$$ (5)

where ε₀ is the vacuum permittivity (8.854 pF·m⁻¹) and ε_r is the relative permittivity (BaTiO₃: 1700, ZnO: 8.5, KNN: 500, PZT: 1300). The results were 0.008, 0.186, 0.025, and 0.018 V·m·N⁻¹, respectively. The critical stress (σ_critical) is the minimum mechanical pressure at which a piezoelectric material produces a given threshold voltage:

$${\mathrm{\sigma }}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}=\frac{|{\mathrm{E}}^{\mathrm{o}}|}{{\mathrm{g}}_{33}·\mathrm{t}}$$ (6)

where E^o is the standard reduction potential for ROS and t is the average thickness of four piezoelectric material particles measured from atomic force microscopy (AFM) height profiles acquired during Kelvin probe force microscopy (KPFM) (1.0E-06 m, Fig. S3). The main particle-associated DLS peak diameters averaged approximately 1 μm across the four materials (Fig. S4), supporting the use of this value as a representative length scale in the calculations. The σ_critical values to reach −0.33 V, the standard reduction potential of O₂/·O₂^– (−0.33 V vs. NHE), for ·O₂^– formation were 40.1, 1.8, 13.2, and 18.1 MPa; for ¹O₂ were 97.0, 4.3, 32.0, and 43.8 MPa; and for ·OH were 339.9, 15.1, 111.7, and 153.5 MPa (Fig. 1(d)). The corresponding open-circuit potentials for 1 μm particles at the 1.13 GPa compressed-bubble pressure were 9.0, 210.2, 28.3, and 20.3 V for BaTiO₃, ZnO, KNN, and PZT, respectively. A local one-dimensional gas–water–solid model quantified stress transmission from the compressed bubble to the solids. The stress-transfer calculation used the compressed-bubble state at 1.13 GPa obtained from the bubble-collapse calculation as its initial condition. The solid thickness followed the AFM-based average of 1 μm used in Eqn. (6), with DLS providing an independent representative particle-size scale. The calculated peak surface normal stresses ranged from 1.078 to 1.090 GPa for direct gas–solid contact to 1.744–1.808 GPa for a 100 nm water gap (Fig. S5). The stress histories at the front surface, midplane, and back surface show wave propagation and reflection within the solids (Fig. S6). The depth–time maps show that the internal stress varies with both position and time (Fig. S7). The water–solid impedance mismatch produces a positive pressure reflection that amplifies the interface stress above the incident water-wave pressure. This reflection explains why the calculated peak solid stress can exceed the initial gas pressure. The calculated peak surface normal stresses were more than three times the largest σ_critical value in Fig. 1(d) (339.9 MPa).

2.2. Surface Electrochemistry: Mechano-Electrical energy storage at the interface

Surface electrochemical states of all the four powders before and after 120 min of 40 kHz US in deionized (DI) water were compared to examine if ROS generation requires actual charge exchange at the water–ceramic interface. Contact potential differences (CPD) were compared before and after 120 min US in Fig. S8. The CPD was measured between a conductive AFM tip and the sample. The surface potential distribution reflects residual piezoelectric voltage effects and work function changes from interfacial charge rearrangement [36]. Average CPD values from line profiles extracted from KPFM surface potential distribution of each material shifted as shown in Fig. 2(a). The observed CPD shifts, which represent the net energy stored by mechanical activation and are independent of absolute tip calibration, exceeded the thermal equilibrium energy of electrons at 298 K (25 mV) by 5–12 times [37] (Fig. S8). The CPD shifts persisted after US, indicating that the observed potential changes were not simple transient polarization signals but residual states fixed by charge rearrangement and/or trapped charges at the water–ceramic interface. These CPD shifts provided substantial thermodynamic driving force for redox reactions at the water–ceramic interface. The materials stored energy in a manner analogous to capacitive charging.

Fig. 2.

(a) Average CPD difference and (b) Surface charge density difference of BaTiO₃, ZnO, KNN, and PZT before and after US, and (c) Schematic of modified surface electrical states under mechanical actuation.

Surface charge accumulation capacity was compared before and after US to confirm changes in surface electrical states and residual levels. Cyclic voltammetry (CV) was applied to BaTiO₃, ZnO, KNN, and PZT. The accumulated charge was decreased by 1.30E-02 times in KNN, while it was increased in BaTiO₃ and PZT by 3.96 and 1.52 times, respectively (Fig. 2(b)). ZnO showed a marginal decrease (x0.91). Significant reduction in interfacial charge accumulation capacity of KNN can be from the surface reconstruction of the material due to US. K⁺ and Na⁺ ions may have leached out during this process. Structural and compositional changes under mechanical activation were observed in prior studies [38], [39]. Piezoelectric ZnO/CdS used for dye degradation under US showed progressive decline in degradation efficiency over repeated cycles, with X-ray photoelectron spectroscopy (XPS) confirming concurrent reductions in O–H, Zn–O, and S 2p_1/2,_3/2 binding energies [38]. BaTiO₃ subjected to ball milling for thiol oxidation exhibited narrower distribution of particle sizes, morphological changes accompanied by recrystallization, and a substantial drop in product yield over successive cycles [40].

The absolute magnitude of increased charge density in BaTiO₃ and PZT after US was much higher than those expected from simple electrical double layer (EDL) capacitance charging alone [41]. EDL capacitance from simple capacitive charging is approximately 2 to 4E-05C·cm⁻² [42]. BaTiO₃ and PZT exhibited approximately 32 and 55 times higher values than the EDL upper limit after US, respectively. ZnO also showed approximately 7 times higher value. This could be explained by polarization and interfacial capacitance charging–discharging across the entire potential window. The measured charge exceeds the estimated EDL contribution and is consistent with additional charge-storage processes. The piezoelectric materials could form and maintain (or amplify) modified electrical states on their surfaces through mechanical actuation (Fig. 2(c)).

2.3. Irreversible structural changes

X-ray diffraction (XRD) analysis showed changes in major diffraction peaks on all samples after US (Fig. 3(a)). The corresponding lattice spacing changes (Δd/d₀) were monitored as shown in Fig. 3(b–e). BaTiO₃ and ZnO showed lattice contraction of − 0.10% to − 0.30% at most peaks. KNN exhibited similar average lattice contraction. PZT showed the largest structural deformation with lattice spacing decreased up to − 0.44% at low angle peaks. The full width at half maximum (FWHM) was used to calculate microstrain and confirm the extent of internal crystal deformation or defect accumulation during 120 min of irradiation:

$$\mathrm{\beta }\mathrm{c}\mathrm{o}\mathrm{s}\theta =\frac{\mathrm{k}\mathrm{\lambda }}{\mathrm{D}}+4\mathrm{\epsilon }\mathrm{s}\mathrm{i}\mathrm{n}\theta$$ (7)

where β is FWHM (rad), θ is Bragg angle, k is the shape factor reflecting crystallite morphology (0.9), λ is X-ray wavelength (1.54 Å), D is the effective crystallite size causing X-ray interference, and ε is the microstrain within the crystal. BaTiO₃ showed FWHM increase (ΔFWHM/FWHM₀ ≈ +1.1–2.4%) at 2θ of 22.2–45.4°, corresponding to microstrain ε of 1.2–2.5E-03. ZnO showed FWHM decrease (ΔFWHM/FWHM₀ ≈ −5.6 to − 18.2%) at 2θ of 31.8–36.3°, which can be interpreted as instantaneous local annealing caused by cavitation temperatures of 2000–4000 K [43]. KNN also showed FWHM decrease (−0.5 to − 6%) at 2θ of 22.3–45.3°. PZT showed FWHM increase of + 6–9% at 2θ of 20.8–28.3° with microstrain ε of 3.0–5.0E-03, which was the largest among the four materials.

Fig. 3.

(a) XRD patterns and (b–e) Δd/d₀ of BaTiO₃, ZnO, KNN, and PZT before and after 120 min US.

The magnitude and direction of microstrain observed in XRD were compared with pressure-dependent structural calculations using density functional theory (DFT). DFT calculations at a hydrostatic pressure equal to the 1.13 GPa compressed-bubble pressure showed that BaTiO₃ has normalized unit cell volume V/V₀ of 0.996 under 1.13 GPa pressure, corresponding to Δd/d₀ of − 0.134%. This is like the average Δd/d₀ of − 0.143% measured by XRD. ZnO and PZT exhibited similar results. For KNN, DFT calculations showed abrupt volume reduction to V/V₀ of 0.52 only at 10 GPa, a phase transition threshold about nine times the 1.13 GPa compressed-bubble pressure and above the calculated peak surface normal stresses of 1.078–1.808 GPa. XRD reflects the bulk-averaged lattice response, whereas scanning electron microscopy (SEM) shows the particle surfaces. The SEM images showed retained particle sizes with fractured faces and surface flakes in BaTiO₃, agglomeration in ZnO, an unchanged cuboidal morphology in KNN, and a retained lamellar agglomerate texture in PZT after US (Fig. S9). These observations, together with the XPS compositional changes, are consistent with degradation concentrated near the particle surfaces. The severe electrochemical degradation of KNN without a visible morphological change is consistent with surface reconstruction rather than bulk collapse.

These pressure-dependent structural changes were accompanied by changes in TC degradation kinetics. TC degradation with all the four materials showed distinct elbow shapes in Fig. 4(a). US alone at 40 kHz decreased the TC concentration by only 0.7% after 120 min and 1.9% after 480 min (Fig. S10). Clear distinctions between the pseudo-first-order rate constants of the initial (k₁) and later (k₂) phases, separated by the elbow point (t*), were observed. The decrease in k₂ beyond the elbow point indicates declining reactivity during continued US.

Fig. 4.

(a) TC degradation kinetics and (b) TC degradation over five cycles.

Five Cycles of TC degradation were performed in series. All four piezoelectric ceramics exhibited a decrease in degradation efficiencies from the first to the second cycle as shown in Fig. 4(b) (BaTiO₃: 42.0%→21.1%, ZnO: 47.3%→3.9%, KNN: 51.9%→3.6%, PZT: 67.1%→4.6%). The TC degradation efficiencies with all four materials converged to 1.6–2.9% by the fifth cycle. This irreversible decline was difficult to explain by intermediate scavenging alone. The structure, defects, surface chemistry, and interfacial electrochemical states of piezoelectric materials were changed during the reaction, preventing regeneration to initial driving states. Irreversible changes in material state were accumulated simultaneously, causing rapid decline in driving force in the repeated use.

2.4. Changes in interfacial electrochemical state

Mott–Schottky and electrochemical impedance spectroscopy (EIS) analyses were performed before and after US to analyze the energy state at the interface. BaTiO₃, ZnO, and PZT exhibited flat-band potential (V_fb) shifts of − 0.11, +0.58, and + 0.19 V in Fig. 5, respectively, indicating irreversible alteration of the interfacial energy state. KNN showed V_fb sign reversal from − 1.35 to + 0.73 V, suggesting a fundamental transition from n-type to p-type surface (Table S1). EIS results indicated material-dependent changes in charge transfer resistance (R_ct). BaTiO₃ and ZnO showed R_ct decreases of 29.7% and 1.3%, respectively, suggesting partially facilitated charge transfer from newly exposed surfaces. PZT showed modest R_ct increase (1.14 times). In contrast, KNN exhibited R_ct surge by ∼ 3,040 times (269.1 to 819,143 Ω), confirming complete loss of interfacial charge transfer capability.

Fig. 5.

Mott–Schottky and EIS plots before and after 120 min US: (a, b) BaTiO₃, (c, d) ZnO, (e, f) KNN, and (g, h) PZT.

XPS analysis identified preferential dissolution of the large, network-modifying cation (the A-site cation in ABO₃ perovskites, and the Zn cation in wurtzite ZnO) as the common origin of these changes: Ba²⁺ in BaTiO₃ (Ti_2p/Ba_3d ratio + 6.6%), Zn²⁺ in ZnO (O_1s/Zn_2p ratio × 3.00), and K⁺ in KNN (K_2p/Nb_3d ratio − 17.6%). PZT showed minimal compositional variation (Fig. S11). This leaching was quantified directly by inductively coupled plasma optical emission spectroscopy (ICP-OES) of the supernatants after 120 min in the TC solution (Table S2). Under 40 kHz US, this cation release was 2.3–3.6 times that under stirring alone, while B-site elements remained below the limit of quantitation. Pb from PZT was undetectable under stirring but reached 1.04 mg·L⁻¹ under US, indicating that the minimal compositional variation observed by XPS reflects a small fractional loss of Pb rather than its absence. The initial TC amount was 20.3 μmol in each run, and the first-cycle removal corresponded to approximately 8.5, 9.6, 10.5, and 13.6 μmol of TC for BaTiO₃, ZnO, KNN, and PZT, respectively. The cation release under US corresponded to 23.6–29.9 μmol of Ba, 79.4–89.9 μmol of Zn, and 39.3–47.6 μmol of K, exceeding the removed TC by 2.8–9.4 times on a molar basis. Pb release from PZT corresponded to 0.14–1.51 μmol, or 0.01–0.11 times the removed TC. The material loss of BaTiO₃, ZnO, and KNN was therefore comparable to or greater than the TC removal, whereas PZT showed limited Pb loss relative to TC removal. The released cations corresponded to equivalent fully depleted surface layers of approximately 3.1–3.9 nm for Ba in BaTiO₃, 3.6–4.1 nm for Zn in ZnO, 7.6–9.3 nm for K in KNN, and 0.03–0.27 nm for Pb in PZT for 1 μm particles. These thicknesses were estimated using Eqn. (8):

$$f=1-{\left(1-2\delta /d\right)}^3$$ (8)

where f is the released fraction of each cation, δ is the depleted thickness, and d is the particle size. The depleted layers are thin relative to the particle sizes of approximately 0.3–1.1 μm observed by SEM and the main DLS peak diameters of approximately 1 μm (Figs. S4 and S9). The consumed fraction scales approximately as 6δ/d for a fixed depleted thickness. Smaller particles with higher specific surface areas would therefore lose a larger fraction of their cations under the same cavitation exposure. Eqn. (6) indicates that σ_critical is inversely proportional to the particle thickness t. Smaller particles would therefore require higher stresses to reach the standard reduction potentials for ROS formation. Preferential K dissolution can change the stoichiometry of KNN in aqueous media [44], and its electrical properties are sensitive to alkali composition [45]. The greater K release and the decrease in the surface K/Nb ratio are therefore consistent with the more pronounced interfacial electrochemical changes in KNN compared with PZT. A-site leaching created vacancies that initially reset band bending and expanded reactive surface area, but this process is irreversible and cumulative. The progressive structural degradation may have contributed to the transition from k₁ to k₂ at each elbow point (t* = 20–60 min) in Fig. 4(a). K⁺ ions in KNN were selectively leached. The Nb framework corresponding to the B-site was relatively maintained and the surface was reconstructed into a K-deficient/Nb-excess layer with broken stoichiometric ratio. This led to insulation of the KNN surface: K⁺ leaching leaves vacancies with effective charge of − 1 [46]. The formed vacancies acted as electron traps or hole acceptors. Consequently, K⁺ vacancies with an effective negative charge trapped free electrons, depleting mobile carriers and transforming KNN into an insulator [47]. This is consistent with the observed V_fb sign reversal and R_ct surge by ∼ 3,040 times.

Possible contributors to A-site leaching in BaTiO₃, KNN, and PZT include electron collisions in cavitation plasma and surface bond weakening by ROS generated through the piezoelectric effect [35], [48]. The gas temperature of 3,751 K used in the Maxwell–Boltzmann analysis can provide 31.18 kJ·mol⁻¹ of thermal energy, and the 1.13 GPa compressed-bubble pressure can provide 5.29 kJ·mol⁻¹ of mechanical energy through Eqn. (2). The equilibrium Maxwell–Boltzmann reference at 3,751 K was used to compare electron fractions above the A-site bond-energy criteria (Fig. S12(a)). The larger reference fraction for KNN is consistent with its lower A-site bond-energy criterion. Bonds other than A-site in BaTiO₃, KNN, and PZT materials with strong binding energies resist leaching. This is consistent with the XPS results showing strong leaching only from A-site materials. The irreversible structural and interfacial electrochemical changes support describing the observed behavior as finite mechano-electrochemical energy conversion accompanied by material degradation and loss of function.

2.5. Implication

The energy-conversion role and finite functional lifetime of piezoelectric materials should guide their application design. Rather than maximizing stress to increase voltage output, the design should consider: (i) the voltage required for a specific application, (ii) the stress range that prevents material degradation, (iii) material-specific piezoelectric coefficients, and (iv) bond stability under the thermal, plasma, and ROS conditions of the operating environment. Fig. 6 maps the mechanical stresses relevant to piezoelectric applications into three operational zones. Within Zone 1 (≤100 MPa), voltage generation scales linearly with material-specific g₃₃: BaTiO₃ generates 0.08 mV, ZnO 1.86 mV, KNN 0.25 mV, and PZT 0.18 mV at 10 kPa. The tens to hundreds of mV required for sensors and energy harvesting are achievable at stress levels of kPa to several MPa without entering degradation regimes. Applications requiring stable and repeated operation should be designed within the Zone 1.

Fig. 6.

Classification of mechanical stress zones (Zone 1–3) and corresponding piezoelectric potentials.

The Zone 2 (100 MPa–1 GPa) enables higher voltage output, but it is the region where structural degradation is initiated. This zone encompasses local impact contacts and mechanical press conditions. Protective coatings and stress-buffering structures require verification of their durability under repeated cavitation. Polymer coatings may introduce a secondary contamination risk through fragmentation, as ultrasound can generate microplastics from bulk polymers in water [49]. Coating cracking and delamination can also expose the underlying material to direct cavitation erosion [50]. Composite components may provide mechanical support or additional charge-transfer and reaction pathways [15], [51], but these functions do not establish that the piezoelectric surface is protected from erosion or leaching. Nanostructuring changes surface area and stress distribution but does not itself ensure durability, as piezoelectric nanorods can undergo cavitation-induced fracture [52]. The extension of the sacrificial transducer model to these architectures should depend on whether sustained redox output is accompanied by irreversible degradation of the piezoelectric phase. Durability assessment should track structural and compositional changes, ion and particle release, and activity retention during repeated operation.

The Zone 3 (≥1 GPa) is where phase transitions, cracking, and fracture of typical ceramics can occur. The 1.13 GPa bubble internal pressure used in the stress-transfer calculation falls within this zone. Static application of GPa-level stress to bulk ceramics is impractical, but cavitation collapse can deliver such stresses as short-duration local events. The pressure required to generate ROS from water compression alone (2.8–8.1 GPa) falls within this zone. Under US cavitation, selective weakening of A-site bonds by the combined thermal-plasma-ROS environment, leading to ion leaching should be concerned. This can release hazardous ions, such as Pb²⁺, and cause secondary contamination that is directly in conflict with the goals of environmental remediation. The recovered BaTiO₃ powder showed fractured faces and surface flakes, and the recovered ZnO powder appeared agglomerated after US (Fig. S9). These morphological changes and the activity decline over five cycles (Fig. 4(b)) limit the reuse of unprotected powders. Unprotected applications of ceramic piezoelectric powders in US-driven environmental processes should be avoided.

2.6. Conclusion

Piezoelectric ceramics under 40 kHz US cavitation undergo irreversible structural and electrochemical changes. The local transmission model quantifies the stress delivered to the solids using the compressed-bubble state at 1.13 GPa obtained from the bubble-collapse calculation as its initial condition. Lattice contraction, strain accumulation, CPD shifts, and irreversible V_fb shifts collectively indicate that cavitation permanently alters both crystal structure and interfacial energy states. These findings challenge the prevailing interpretation of piezocatalysis. The non-steady-state kinetics, pronounced activity decline over repeated cycles, and A-site cation release indicate limited functional stability under the tested cavitation conditions. The driving force for the interfacial redox chemistry is supplied by cavitation, not by ultrasonic vibration, which alone is too weak to polarize ceramic. Cavitation plasma and ROS provide possible additional routes for weakening A-site metal–oxygen bonds. These findings support describing the tested powders as finite mechano-electrochemical transducers whose operation is accompanied by irreversible structural and electrochemical changes. This description concerns their energy-conversion role and functional lifetime and does not exclude catalytic steps at the evolving solid–water interface. BaTiO₃, ZnO, and KNN also acted as sacrificial materials, with cation release exceeding the removed TC on a molar basis, whereas PZT lost activity with limited Pb release. In mechanically driven environmental redox systems, engineered defects and high-stress activation can enhance apparent activity while also creating weak metal–oxygen bonds susceptible to selective leaching. Future piezoelectric material design should therefore report redox output together with the allowable operating stress range, material lifetime, recoverability, and ion-release risk.

3. Materials and methods

3.1. Materials

BaTiO₃ powder (Sigma-Aldrich, USA) and ZnO powder (Sigma-Aldrich, USA) were used. Potassium carbonate (K₂CO₃, Sigma-Aldrich, USA), sodium carbonate (Na₂CO₃, Sigma-Aldrich, USA), niobium(V) oxide (Nb₂O₅, Sigma-Aldrich, USA), oxalic acid (Sigma-Aldrich, USA), lead(II) nitrate (Pb(NO₃)₂, Sigma-Aldrich, USA), zirconium(IV) sulfate tetrahydrate (Zr(SO₄)₂·4H₂O, Sigma-Aldrich, USA), titanium(IV) isopropoxide (TTIP, Ti(OiPr)₄, Sigma-Aldrich, USA), acetylacetone (AcAc, Sigma-Aldrich, USA), ethanol (Sigma-Aldrich, USA), methanol (Daejung, Korea), acetonitrile (ACN, Daejung, Korea), nitric acid (HNO₃, Sigma-Aldrich, USA), ammonium hydroxide (NH₄OH, Daejung, Korea), and sodium sulfate (Na₂SO₄, Sigma-Aldrich, USA) were used. DI water was produced using AquaMA-Ultra370 (Young In Chromass, Korea). All reagents were used without purification.

3.2. Synthesis of KNN

A modified oxalate-precursor co-precipitation method was employed to synthesize 10 g of KNN ceramics. First, 7.73 g of Nb₂O₅ powder was activated by calcination with excess carbonates at 900 °C and subsequent acid treatment (pH 1) to yield reactive niobic acid precipitate. This precipitate was dissolved in an aqueous solution containing 22.00 g of oxalic acid dihydrate (molar ratio of Nb to oxalic acid = 1:3) at pH 6–7 to form a stable ammonium niobium oxalate complex. Separately, an alkali nitrate solution was prepared by dissolving 2.09 g of K₂CO₃ and 1.60 g of Na₂CO₃ (including 4 mol% excess to compensate for volatilization) in dilute HNO₃. The alkali solution was slowly added to the niobium precursor solution, and the mixture was co-precipitated by adjusting the pH to 8 using NH₄OH. The resulting suspension was dried at 120 °C and calcined at 650 °C for 4 h. Finally, the calcined powders were sintered at 1100 °C for 2 h in a closed alumina crucible with atmospheric powder to suppress alkali volatilization.

3.3. Synthesis of PZT

A modified co-precipitation method was employed using separate aqueous and organic precursor solutions to prevent premature hydrolysis to synthesize approximately 10 g of PZT powder. Solution A was prepared by dissolving 10.17 g of Pb(NO₃)₂ and 5.68 g of Zr(SO₄)₂·4H₂O in 400 mL of DI water containing 2.0 mL of 70% HNO₃ to prevent premature hydrolysis. Simultaneously, Solution B was prepared by stabilizing 4.19 g of TTIP with 1.47 g of AcAc in 100 mL of ethanol under stirring until transparent yellow solution was obtained. Both solutions were simultaneously added dropwise into a base bath of 500 mL DI water adjusted to pH 9 using NH₄OH under vigorous stirring. The resulting white precipitate was aged for 2 h while maintaining the pH. Subsequently, the precipitates were filtered and extensively washed with DI water and ethanol to remove residual sulfate and nitrate ions. Finally, the precursor was dried at 120 °C overnight and calcined at 850 °C for 2–4 h.

3.4. Characterization

The crystal structures of the synthesized powders were analyzed using an X-ray diffractometer (D8 ADVANCE, Bruker, Germany) with Cu Kα radiation (λ = 1.5418 Å). The diffraction patterns were recorded in the 2θ range of 20°–80° with a step size of 0.02° and a scan speed of 0.6 s/step.

The powders for XRD and SEM analyses were treated with US in DI water. The powders for XRD analysis were recovered after ultrasonic treatment by centrifugation at 9,000 rpm for 15 min. The supernatant was discarded, and the recovered powders were washed five times with a 1:1 methanol–water mixture in a 15 mL conical tube. Each washing step involved gentle manual shaking for approximately 10 min without vigorous shaking or vortex mixing. The washed powders were dried at 80 °C for 12 h. No additional grinding was performed before XRD analysis to avoid mechanically induced changes. SEM images of the powders before and after US were acquired using a field-emission SEM (SU5000, Hitachi, Japan) at an accelerating voltage of 15.0 kV after Pt sputter coating at 15 mA for 60 s.

The surface chemical states and elemental compositions were investigated using XPS (Quantum2000, Ulvac-PHI, Japan) equipped with a monochromatic Al Kα X-ray source (1486.6 eV, 25 W, 15 kV). Survey spectra were acquired with a pass energy of 117.4 eV, while high-resolution spectra for O_1s, Ti_2p, Zn_2p, Nb_3d, Pb_4f, etc., were collected with a pass energy of 58.7 eV. The X-ray beam size was set to 100 µm and charge neutralization was applied during measurements.

Surface potential and piezoelectric response measurements were conducted using an AFM (NX10-AFM, Park Systems, Korea). KPFM measurements were performed in KPFM mode with a scan rate of 0.3–0.5 Hz, setpoint of 18.5–19.5 nm, drive voltage of 1.0 V, and gain of 0.02–0.05. Work function offset was set to 0.5 eV without calibration. PFM measurements were performed in contact mode using a Pt/Ir-coated conductive silicon tip (PPP-EFM, spring constant ∼ 2.8 N∙m⁻¹). An alternating-current (AC) voltage of 1–5 V was applied at contact resonance frequency (300–350 kHz) to enhance sensitivity. The scan rate was set to 0.5–1.0 Hz with a setpoint force of 10–100 nN under ambient condition.

Cavitation bubble size was evaluated by DLS using a Zetasizer (Nano ZS90, Malvern, UK). DI water (1 mL) was loaded into a dip cell and subjected to US for 10 min using an ultrasonic bath (Powersonic 405, 40 kHz, manufacturer-specified electrical power consumption of 350 W, Hwasin Tech., Korea). The measurements were conducted immediately after the cessation of ultrasound irradiation to characterize the nascent bubbles.

Electrochemical measurements were performed using a potentiostat (VersaSTAT4, AMETEK, PA, USA) in a standard three-electrode system using a 0.5 M Na₂SO₄ aqueous solution as the electrolyte. A platinum wire and an Ag/AgCl electrode were used as the counter and reference electrodes, respectively. The working electrode was fabricated by coating 20 mg of the piezoelectric powder onto a 12 cm² fluorine-doped tin oxide glass substrate. CV was performed in the potential range of − 1.0 to 1.0 V vs. Ag/AgCl at a scan rate of 50 mV∙s⁻¹. Mott–Schottky analysis was conducted at a frequency of 1 kHz with an AC amplitude of 10 mV. EIS measurements were performed over a frequency range of 0.1 Hz to 100 kHz with an AC amplitude of 10 mV at open circuit potential.

KNN and PZT were synthesized in-house, and the stoichiometries reported here (K_0.5Na_0.5NbO₃ and PbZr_0.53Ti_0.47O₃) were determined by ICP-OES (Optima 8000, PerkinElmer, Waltham, Massachusetts, USA) after acid digestion of the as-prepared powders, whereas BaTiO₃ and ZnO were commercial powders used as received. Dissolved constituent metals in the supernatants were quantified by ICP-OES after 0.22-µm filtration and acidification to 2% (v/v) HNO₃. Suspensions of the four piezoelectric materials (1.0 g·L⁻¹ in 300 mL of 30 mg·L⁻¹ TC solution) were sampled after 120 min of 30 rpm stirring and after 120 min of 40 kHz ultrasonic treatment.

3.5. Determination of the piezoelectric coefficient

The d₃₃ values were determined from the PFM amplitude-voltage butterfly loops using:

$${\mathrm{d}}_{33}=({\mathrm{A}}_{\mathrm{m}\mathrm{a}\mathrm{x},\mathrm{f}\mathrm{w}\mathrm{d}}+{\mathrm{A}}_{\mathrm{m}\mathrm{a}\mathrm{x},\mathrm{b}\mathrm{w}\mathrm{d}})/(2\times {\mathrm{V}}_{\mathrm{A}\mathrm{C}})$$ (9)

where A_max,fwd and A_max,bwd are the maximum piezoresponse amplitudes at the butterfly loop wing peaks from the forward and backward sweeps, respectively. V_AC is the AC driving voltage applied in the PFM measurement (1.5 V). The unit equivalence pm∙V⁻¹ = pC∙N⁻¹ holds.

3.6. TC degradation experiments

The piezoelectric degradation of TC was conducted in an ultrasonic bath (Powersonic 405, Hwasin Tech., Korea) operating at a frequency of 40 kHz. The manufacturer-specified electrical power consumption was 350 W. The rectangular tank had internal dimensions of 300 × 155 × 150 mm and a capacity of 5.7 L. The generator was built into the bath, and five ultrasonic transducers were mounted on the tank bottom. A 600 mL tall-form quartz beaker was placed at the center of the tank bottom, directly above the central transducer. The bath was filled with approximately 3 L of DI water so that the bath-water level matched the solution level in the beaker. The reaction temperature was maintained at 25 °C with cooling. The DI bath water and TC solution were used without degassing.

The synthesized piezoelectric powder (300 mg) was dispersed in 300 mL of the TC aqueous solution (initial concentration: 30 mg∙L⁻¹).

A control experiment was conducted under identical conditions without piezoelectric powder to quantify direct sonolytic degradation of TC. The suspensions containing piezoelectric powder were subjected to ultrasonic irradiation for 120 min in the dark to avoid any photocatalytic interference. The powder-free control was extended to 480 min to evaluate cumulative TC removal by US alone.

The particle size distributions of the four powders in water before and after US were measured by DLS using the Zetasizer. Separated peaks at 5.3–5.6 μm in BaTiO₃, ZnO, and PZT, which accounted for 11.2–30.7% of the scattered intensity, were excluded from the distributions in Fig. S4 as bubble contributions. These large-size populations may also contain particle agglomerates.

A separate heating test was conducted with the lid closed and both the cooling system and bath heater switched off. The TC solution and surrounding DI bath water both increased from 25 °C to approximately 50 °C during 60 min of US. The average net heating rate of the reaction solution was calculated using Eqn. (10):

$$P_{\mathit{net}}=\frac{m{c}_p\mathrm{\Delta }T}{\mathrm{\Delta }t}$$ (10)

where m = 0.300 kg, c_p = 4.18 kJ·kg⁻¹·K⁻¹, ΔT = 25 K, and Δt = 60 min. The temperature rise of the 300 mL reaction solution corresponded to an average net heating rate of approximately 8.7 W. This value includes heat exchange with the surrounding bath and represents net heating over the measurement interval.

The residual concentration of TC was quantified using a high-performance liquid chromatograph (HPLC, Younglin 9100, South Korea) equipped with a diode array detector. Sample separation was performed on a Sunfire C18 column (4.6 × 260 mm, 5 μm, Waters, USA) maintained at 30 °C. The mobile phase consisted of methanol, ACN, and water containing 0.03 M oxalic acid in a volume ratio of 20:20:60 (v/v/v). The mineralization efficiency was assessed by monitoring the total organic carbon (TOC) removal using a DRB200 digital reactor (Hach, USA). The degradation experiment was repeated for 5 cycles under identical conditions. The piezomaterials were collected after each cycle via centrifugation, washed three times with DI water and methanol, and dried for 12 h before the next run.

3.7. Computational details

First-principles calculations were performed based on DFT using the Quantum ESPRESSO package (v6.4.1) with the generalized gradient approximation parameterized by the Perdew-Burke-Ernzerhof functional [53]. The ion–electron interactions were described using projector augmented wave pseudopotentials obtained from the PSLibrary [54]. To simulate the structural response under high-pressure conditions, supercell models containing 40 atoms were constructed for BaTiO₃, ZnO, KNN, and PZT, where the Brillouin zone was sampled using an optimized Monkhorst-Pack k-point mesh. The plane-wave kinetic energy cutoff and charge density cutoff were set to 70 Ry and 560 Ry, respectively, ensuring high precision in the stress tensor calculations. Geometry optimizations were conducted using the Broyden–Fletcher–Goldfarb–Shanno algorithm for both ion and cell dynamics, with strict convergence criteria set to 1.0E-05 Ry for total energy, 3.8E-04 Ry∙Bohr^-1 for interatomic forces, and 0.5 kbar for pressure [55]. Variable-cell relaxation calculations were performed to model the piezoelectric response under cavitation-induced mechanical stress at hydrostatic pressures of 0–10 GPa, allowing for the full relaxation of lattice parameters and atomic positions until the target stress conditions were met.

The thermodynamic feasibility of ROS generation via ultrasonic cavitation was evaluated using the Keller–Miksis equation [32]. The initial bubble diameter of 5.56 μm was measured immediately after 40 kHz US. The initial internal pressure was determined using the Young-Laplace equation incorporating atmospheric pressure (101.3 kPa) and the surface tension of water (0.072 N·m⁻¹ at 293 K). The bubble collapse was modeled as an adiabatic process with a polytropic exponent (γ) of 1.4 for diatomic air, and a hard-core radius of R₀/100, where R₀ is the initial bubble radius of 2.78 μm, was included in the gas pressure relation for numerical stability. The driving acoustic pressure of the ultrasonicator was 150 kPa at 40 kHz. The ordinary differential equations were solved using the Radau method with adaptive time-stepping, and the temporal evolution of bubble diameter D(t), internal pressure p_B(t), and temperature T(t) was recorded (Fig. S1). The compressed-bubble state used in the stress-transfer calculation had an internal pressure of 1.13 GPa. The electron-energy analysis used a gas temperature of 3,751 K. The spatial distributions of pressure and temperature for the compressed-bubble state (Fig. 1(a, b)) were visualized with time history data [56].

The local stress-transfer calculation used a one-dimensional gas–water–elastic-solid model initialized at a compressed-gas pressure of 1.13 GPa. The initial bubble diameter of 5.56 μm determined the gas mass. The compressed source radius and density were 0.33236 μm and approximately 1049.5 kg·m⁻³, respectively, from the polytropic relation used in the bubble-collapse calculation. The gas-column length was four-thirds of the compressed radius, preserving the source volume per projected area. The solid thickness was the AFM-based average of 1 μm used in Eqn. (6), supported by the representative DLS particle-size scale. The calculation compared direct gas–solid contact with a 100 nm water gap at the start of pressure release. All phases were initially at rest, and the water and solid were initially at ambient pressure.

The calculation solved continuity of normal traction and velocity at each interface using a Lagrangian finite-volume scheme. The gas used an ideal-gas equation of state with an exponent of 1.4. Water used a stiffened-gas equation of state with an exponent of 7.15, calibrated to a density of 998 kg·m⁻³ and a sound speed of 1480 m·s⁻¹ at ambient pressure. The solids used linear longitudinal elastic relations. The density and longitudinal elastic modulus were 6020 kg·m⁻³ and 162 GPa for BaTiO₃ [57], 5700 kg·m⁻³ and 210.9 GPa for ZnO [58], 4510 kg·m⁻³ and 152.6 GPa for KNN [59], [60], and 7750 kg·m⁻³ and 111 GPa for PZT [61].

The simulation covered 2 n_s with a 2.5 nm near-interface grid and a Courant factor of 0.28. The outer water domains extended 8 μm on each side. Refinement from 5 to 2.5 nm changed the peak surface stress by 0.24–0.27% and the peak thickness-averaged stress by 2.11–2.44% for the 100 nm gap. Halving the Courant factor changed the peak surface stress by less than 0.007%. Doubling the outer water domains gave identical reported stress peaks at the reported precision. These checks establish numerical consistency within the local model. The calculation resolves stress transmission and reflection from the prescribed compressed state. Time is measured from the start of pressure release, and depth is measured from the front surface in the initial solid configuration. Positive normal stress denotes compression, and negative normal stress denotes tension.

Electron energy relaxation was evaluated at a fixed gas temperature of 3,751 K and pressure of 1.13 GPa (Fig. S12(b–d)). The neutral gas contained 79% N₂ and 21% O₂ by number, with an ideal-gas number density of 2.1820E + 28 m⁻³. A dilute test-electron population evolved through isotropic elastic electron–neutral collisions without affecting the neutral gas. The three initial distributions were Maxwell–Boltzmann distributions at 1,000 and 10,000 K and a normalized Gaussian energy distribution centered at 1.5 eV with a standard deviation of 0.25 eV. The latter was truncated at zero energy.

The normalized energy probability density p(E,t) was propagated using the small electron-to-molecule mass-ratio Fokker–Planck approximation to the elastic collision operator. The energy-space flux, probability-density evolution, and energy diffusion coefficient were given by Eqns. (11), (12), (13):

$$J=-D(E)[\frac{\partial p}{\partial E}-p\frac{\partial ln(p_{\mathit{eq}})}{\partial E}]$$ (11)

$$\frac{\partial p}{\partial t}=-\frac{\partial J}{\partial E}$$ (12)

$$D(E)={\mathrm{\Sigma }}_s\frac{2m_e}{M_s}{n}_s{\sigma }_{m,s}(E)v(E)E{\theta }_g$$ (13)

The gas thermal energy was defined as:

$${\theta }_g=k_B{T}_g$$ (14)

where T_g is the gas temperature, θ_g is expressed in eV, p_eq is the Maxwell–Boltzmann energy probability density at T_g, m_e is the electron mass, E is in eV, v(E) is the electron speed, and n_s, M_s, and σ_m,s are the number density, molecular mass, and elastic momentum-transfer cross section of species s. The Phelps N₂ and O₂ cross-section collection was taken from the archived Cantera-format dataset [62]. The elastic momentum-transfer cross section was approximated by subtracting the tabulated excitation cross sections from the effective cross section, with negative values clipped to zero. The stationary distribution of this elastic operator is the Maxwell–Boltzmann distribution at T_g.

The calculation used a conservative implicit finite-volume scheme with zero probability flux at 0 and 12 eV. The final grid contained 2,400 cells with a time step of 0.0625 ps over 500 ps; comparison with 1,200 cells and 0.125 ps changed the settling times for 1% temperature error or 1% threshold-fraction error by no more than 2 ps. Probability conservation error remained below 1.6E-12. Electron temperature was defined from the mean energy using consistent energy units:

$$T_e=\frac{2\langle E\rangle }{3k_B}$$ (15)

where 〈E〉 is the mean electron energy. The maximum relative error (%) in the threshold-exceeding electron fractions was calculated as:

$${\delta }_F={\mathrm{m}\mathrm{a}\mathrm{x}}_{j}100|\frac{F_j}{F_{\mathit{eq},j}}-1|$$ (16)

where F_j denotes the fraction of electrons above criterion j and F_eq,j is the corresponding Maxwell–Boltzmann fraction. The three energy criteria were 31.8, 77.1, and 270.2 kJ·mol⁻¹.

The calculation quantifies relaxation under the specified fixed bath conditions. The Maxwell–Boltzmann stationary state is prescribed by the elastic operator, so convergence to this state quantifies relaxation within the chosen model rather than validates a Maxwellian distribution or equality of electron and gas temperatures during cavitation.

4. Availability of data and materials

The data and materials supporting the findings of this study are available from the corresponding author upon reasonable request.

CRediT authorship contribution statement

Gunn Park: Writing – original draft, Visualization, Validation, Methodology, Conceptualization. Jae-Woo Park: Writing – review & editing, Supervision, Project administration, Funding acquisition.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary data

The following are the Supplementary data to this article:

Supplementary Data 1

Supplementary data on bubble collapse, stress transmission, electron energy relaxation, material characterization, tetracycline degradation, and cation release.