Electrostriction: it is just a phase

Domain: physics
Published: 2025-11-27
Editorial outcome: minor_revision

Abstract
The discovery of “giant” (or non-classical) electrostrictors has reignited interest in electrostriction, a second-order electromechanical coupling phenomenon. While the relationship between the generated strain and the square of the electric field may seem straightforward, determining the sign of the electrostrictive coefficients proves challenging, as evidenced by the opposite signs reported by various research groups. We show that electrostrictive coefficients must be treated as complex values with their sign dictated by their phases rather than by the overall shape of the strain versus electric field curve. Moreover, our analysis of the electrostrictive properties of La2Mo2O9 ceramics reveals that both the electrostrictive coefficients and the induced strains may undergo sign changes when the frequency varies, with each sign change occurring at its own critical frequency. Finally, we introduce a model to explain the differing frequency response of the electrostrictive coefficients.

Introduction
Considering the lead-free requirement for electromechanically active materials, non-classical (so-called “giant”) electrostrictors have attracted significant attention over the past decade1–5. Electrostrictive coefficients define the quadratic relationship between strain and either the electric field or the polarization (x =M EE or x =Q PP, with M and Q denoting the field and polarization electrostrictive tensors, respectively, P the polarization, and E the electric field). The quadratic dependence of the strain on the electric field indicates that the deformation direction is independent of the electric-field polarity.

The sign of the longitudinal tensor component (the electrostrictive “coefficient”) can be either positive or negative. For example, most relaxor ferroelectrics exhibit positive electrostrictive coefficients6–9. On the contrary, several materials have been reported to have negative electrostrictive coefficients, including doped ceria ceramics (CeO2)10–13, La2Ce2O7 ceramics14, stabilized bismite (Bi2O3)15, LAMOX (La2Mo2O9-based materials)16,17, multi-cation stabilized zirconia18, van der Waals ferroelectric CIPS (CuInP2S6)19,20, methylammonium lead triiodide (MAPbI3)21 and relaxor ferroelectric polymers22. The different signs of these electrostrictive coefficients have been intuitively connected to the material’s tendency for tensile or compressive behavior, based on the interatomic potentials underlying a rigid ion model23. Recent ab initio calculations offer an additional explanation for these positive or negative signs. They demonstrated that the electrostrictive response in undoped BaZrO3 (which displays classical electrostriction) is driven by three transverse optical modes. Among these, two contribute positively while the third contributes negatively24. In such materials, the transverse optical modes’ amplitude ultimately determines the sign of the electrostrictive response.

However, the inconsistency in the reported signs of the longitudinal electrostrictive coefficients in the same material is often overlooked. For example, Gd-doped ceria thin films have been reported to exhibit compressive1 or tensile4 electrostrictive strains, attributed respectively to negative and positive electrostrictive coefficients. Similar cases also occur in classical electrostrictors, such as MASD25,26, NaNO227,28, BaF2 and SrF2 29,30.

Two explanations have been proposed to describe the difference in generated electrostrains (rather than in electrostrictive coefficients). The first is that the electric field amplitude affects the sign of the strain31. It was based on the fact that the strain is negative at low excitation amplitude despite the overall hysteretic strain-electric parabola being concave-upward, thereby corresponding to an overall positive strain. This has been interpreted as a change from compressive at low field amplitude (assimilated to a negative electrostrictive coefficient) to tensile at high field amplitude (positive electrostrictive coefficient). Such a behavior would be consistent with the positive strain of Gd-doped ceria thin films under high (50kV/mm) electric field4 and negative strain under low (3.5kV/mm) electric field1. The second explanation recall that the measured quadratic strain is the sum of the electrostrictive strain (x =M⋅E2) and the Maxwell strain (that is always compressive and induced by the electrostatic attraction between the electrodes) xM = − 1/2 ε0 εr S E2 with ε0 and εr the vacuum and relative permittivities, respectively, and S the elastic compliance. The material parameters (permittivity and elastic compliance) depend on the angular frequency ω of E(ω), the applied AC electric field. When 2M≥ε0 εr S, the measured strain will be dominated by the electrostrictive strain (x ≥xM) and reflect its sign, whereas when the Maxwell strain dominates, the measured strain will be compressive regardless of the sign of the electrostrictive coefficient. Maxwell strain effect in dielectric elastomers and ferroelectric polymers is usually taken into account due to their large elastic compliance32,33 (large S). Despite the fact that the Maxwell strain is often overlooked in dielectric ceramics and thin films due to their stiffness (small elastic compliance S), the Maxwell stress induced by strong fields may no longer be negligible due to the potential drastic increase (up to 105-fold) in relative permittivity caused by the hopping of ionic defects under the field34, especially at low frequency (mHz range) or under DC bias. Consequently, the sign of the overall strain may be either negative (i.e., a compression) or positive (i.e., an extension) under different frequencies or electric field strengths because of the (compressive) Maxwell strain potentially overcompensating the tensile electrostrictive strain.

Here, we report an intriguing observation: the sign of the electrostrictive M coefficient of bulk La2Mo2O9 samples changes sign at a given frequency, despite the field amplitude being held constant. This behavior cannot be explained by any of the previously proposed origins. We show that the sign of the strain and the sign of the M coefficient can change at different frequencies. We demonstrate that both sign changes can be explained consistently by considering that the electrostrictive M coefficient is a complex number, using an anharmonic oscillator pair model. The sign is imposed by the frequency-dependent phase of the electrostrictive response. We illustrate this with experimental results on La2Mo2O9 ceramics. The tensile-compressive change occurs between 2 and 3Hz, while the M<0 to M>0 change occurs between 9 and 10Hz. Considering the hystereses in dielectric, elastic, and electrostrictive couplings, we show that only the electrostrictive phase varies significantly. This underscore that the corresponding overall loss changes in an electrostrictor are intrinsically related to the electromechanical coupling, rather than being of dielectric or mechanical origins.

Results
Anharmonic oscillator pair
Electrostriction is a property of all dielectric materials, referring to the mechanical deformation (denoted as δ) that develops proportional to the squared amplitude of the applied electric field. This quadratic (nonlinear) response stems from a force deriving from the anharmonicity of the potential, as the electrostrictive coefficients are third derivatives of the free energy. As a result, the amplitude of the ion shift induced by the external electric field is frequency-dependent, causing relaxation or resonance processes for both dielectric and electromechanical mechanisms. Different dielectric mechanisms (electronic, atomic, dipolar) have their own characteristic frequencies. Generally, as the driving frequency approaches a characteristic frequency, the amplitude of dielectric permittivity or electromechanical response will undergo relaxation or resonance depending on the damping, while the dielectric and electromechanical loss will peak.

To model electrostriction, we consider a pair of anharmonic oscillators between ions of opposite charges and study the deformations induced by the application of an AC electric field. Figure 1 illustrates the case of a one-dimensional crystal lattice. A dipole pair with opposite directions (due to the mirror symmetry) is selected within the dashed boxes, maintaining charge conservation and periodicity. The motion equation of the center of mass of a pair of ions is described as a forced and damped Helmholtz oscillator deriving from an anharmonic potential35:

mδ€ +cδ_ +kδ+aδ2=± q E sin(ωt)

δ€ + m/c δ_ + m/k δ+ m/a δ2=+ q/m E sin(ωt) → δr (t) (1)

δ€ + m/c δ_ + m/k δ+ m/a δ2= − q/m E sin(ωt) → δl (t)

where m is the equivalent mass, k the spring constant, ω0 = √k/m the natural characteristic (intrinsic) frequency, ω the angular frequency of the excitation, c the viscous damping coefficient, and t the time. The plus or minus before qE stems from the scalar product between the electric field (E(ω)=Esin(ωt)) and opposite dipole moments induced by the deformation away from their equilibrium positions of the right (δr(t)) and left (δl(t)) ions, respectively. Therefore, x(t)=δ(t)/r0 =[δr(t)+δl(t)]/r0 corresponds to the strain of the unit cell. The parameter a describes the (cubic) anharmonicity of the single-well potential energy (V(δ)= 1/2kδ2+ 1/3aδ3) responsible for electrostriction. The sign of a determines the extensive (a>0) or compressive (a<0) tendency of the oscillators (see Supplementary information S1).

This model can be applied to entire samples, provided they are homogeneous, or to specific inhomogeneities such as oxygen vacancies near a given element (often a rare-earth dopant in “giant” electrostrictors). In the latter case, the overall response of the sample will be a volume average of these “active” dipoles and the rest of the sample. It is posited that “giant” electrostrictors are based on such active dipoles, the rest of the sample contributing negligibly to the overall response. This latter assumption is supported in rare-earth ceria by the absence of a “giant” response in pure ceria. Therefore, the model we present is not specific to “giant” electrostrictors but encompasses all electrostrictors.

Figure 2a shows the numerical solutions of δ(t) under sinusoidal excitation (see Supplementary information S2 and S3). In the steady-state region, the deformation of the unit cell δ(t) exhibits a frequency doubling relative to the electric field, while the first-order deformations of δr(t) and δl(t) cancel out due to the mirror symmetry.

Figure 2a also illustrates the phase shift of the electrostrictive response as a function of the viscous damping coefficient c. To quantify the inherent dissipation of electromechanical energy conversion under a sinusoidal electric field, we define the dielectric (φelec) and electrostriction (φx) frequency-dependent phases from the linear polarization (with χ*=χe−iφelec the complex electric susceptibility) and of the quadratic electrostrictive strain (with M*=Me−iφx):

E(t)=Re Eei(ωt−π/2) =Esin(ωt)

P(t)=Re ε0 χe−iφelec ⋅Eei(ωt−π/2) =P0 sin(ωt−φelec) (2)

xE (t)=Re Me−iφx ⋅ Eei(ωt−π/2) 2 =x0 sin2(ωt−φx) (3)

where E, P0 and x0 denote the amplitudes of the applied field, polarization, and electrostrictive strain, respectively. The negative sign preceding the phase ensures causality, and φelec and φx take values in [0,π/2) (positive loss tangents) to be physically consistent with energy dissipation. Energy losses are due to factors such as free charge conduction (for φelec) or damping and hopping behavior of oxygen vacancies (for φx).

The expression of measured electrostrictive strain x(t) as a function of the driving electric field frequency is:

x(t)=M(ω)E2sin2[ωt−φx(ω)]−M(ω)E2sin2[φx(ω)] (4)

=xE (t)−xE (0)

The second term ensures that strain is zero at the initial time (t=0). Figure 2c illustrates the deformation-electric field (denoted as δ-E) curves with various damping coefficients for a>0 according to Eq. (4) (see Supplementary information S4 for a<0). Because strain is the deformation divided by a constant (lattice parameter r0), the patterns of δ-E are equivalent to the patterns of strain-electric field (denoted as x-E). The parabola’s shape changes from concave downward (c=0.05) to concave upward (c=22), indicating a sign reversal in the electrostrictive strain response within the material. Note that identical x-E curves can be obtained for various values of a and c (see Fig. S4d in the Supplementary information S4). This suggests that the physical significance of electrostriction may be obscured if only the x-E curves are considered.

The sign of the electrostrictive coefficient has to be redefined to account for all possible electrostrictive responses predicted by the model, rather than simply equating a concave upward x-E parabola (positive strain) with a positive electrostrictive coefficient. The sign of the M coefficient is defined as positive when the maximum of the strain corresponds to the maximum of the electric field. This correspondence holds as long as the maximum strain occurs after the maximum of the electric field and before the next minimum electric field. Inversely, the M coefficient is negative when the maximum of the strain occurs after a minimum of the field and before the next maximum. For example, when c=12 in Fig. 2c, the first extremum of the strain after a maximum of the electric field is a minimum. Therefore, the electrostrictive coefficient is negative, although the x-E parabola is concave upward. The case when c=22 represents the critical point where the sign of M changes from negative to positive. At this point, the phase undergoes a discontinuous jump from π/2 to 0. As the damping coefficient increases further, the x-E parabola becomes concave upward with a positive coefficient M.

This decoupling between the sign of the electrostrictive coefficient and the orientation of the x-E parabola derives from the measurement. Indeed, the x-E curves represented in Fig. 2c correspond to the steady state dynamic response, after about one period (see Fig. 2a), with the initial time set by using the exciting electric field as the reference. As can be seen from Fig. 2a, the average strain is always negative, consistent with the sign of a. However, depending on the phase, the steady-state oscillations are either above or below the deformation at the initial time (see Fig. 2b), resulting in the x-E curves presented in Fig. 2c. Note that this definition of the sign is consistent with existing electrostrictors: the strain-time responses of 0.9Pb(Mg1/3Nb2/3)O3-0.1PbTiO3 (PMN-10PT) as a function of different frequencies are analogous to the pattern in Fig. S5 (see Supplementary information S5).

“Giant” electrostrictors
In addition to ceria-based materials, the LAMOX family (i.e., based on the La2Mo2O9 compound) encompasses several “giant” electrostrictors36. Here, we demonstrate experimentally the frequency-dependent sign change of the overall shape of the x-E curves on bulk (ceramics) of pure La2Mo2O9, “giant” electrostrictor.

Figure 3a shows the measured strain and polarization curves under an applied field of Emax=5kV/mm for frequencies between 0.4 and 100Hz. A drastic change in the electrostrictive strain response is observed in this frequency range, as illustrated by the difference in time evolution of the electrostrictive strain in Fig. 3b (10Hz, extension) and Fig. 3c (0.4Hz, compression). The measured x-E curves shown in Fig. 3a can be compared to the calculated ones that are reported in Fig. 2c. The x-E curve measured at 0.4Hz matches the calculated one for c=3 and a negative M coefficient. The “∞”-shape x-E curve at 3Hz aligns with the pattern calculated for c=6.7, corresponding to the phase φx crossing the π/4 value. The hysteretic upward parabolas in Fig. 3a from 4Hz to 9Hz match the pattern c=12, where the M coefficient remains negative and the phase varies between π/4 and π/2. Finally, the measured non-hysteretic strain curve at 10Hz aligns with the pattern calculated for c=22. This case corresponds to the c value (in the model) and the frequency (in the experiments) where the M coefficient changes sign. Above 10Hz, the experimental x-E curves correspond to the calculated pattern for c=48. Therefore, the experimental electrostrictive x-E curves demonstrate that the M coefficient can change sign between 9 and 10Hz.

Turning to the overall shape of the electrostrictive x-E curves in Fig. 3a, the concave upward shapes above 3Hz (corresponding to extension) turn into concave downward shapes below 2Hz. Therefore, there are two distinct critical frequencies at which a sign change occurs, one for the strain (at 3Hz) and one for the M coefficient (at 10Hz). Each of these critical frequencies corresponds to a specific value of the electrostrictive phase φx: φx =π/4 when the strain changes sign (at 3Hz), whereas φx jumps from π/2 to 0 when the M coefficient changes sign (at 10Hz). These frequencies have been confirmed to be independent of the excitation profile by using triangular and reverse sinusoidal excitations (see Supplementary information S6).

Since electrostriction is an electromechanical coupling, the sign change of the strain and/or the M coefficient could be a consequence of frequency-dependent changes in the dielectric or mechanical properties. Unlike the large variations of φx that are responsible for the frequency-evolution of the electrostrictive properties (Fig. 3), the electric phase φelec (illustrated at 1Hz in Fig. 4a) remains nearly constant across the investigated frequencies (see Fig. 4b) and the permittivity decreases monotonically. The dielectric properties of the La2Mo2O9 ceramics are therefore not the reason behind the frequency-evolution of the electrostrictive properties.

Then, the frequency-evolution of the mechanical properties has been investigated by nanoindentation, a well-established technique for examining quasi-static and dynamic elastic moduli. It confirms the anelastic response of La2Mo2O9 ceramics (see Fig. S11). The mechanical phase (φmec) between the strain (x) and the applied mechanical stress (X) is defined from:

X(t)=X0 sin(ωt)

x(t)=x0 sin(ωt−φmec) (5)

where X0 is the amplitude of the applied stress. The mechanical loss tangent is denoted as tan(φmec).

The measurement of the dynamic elastic modulus (between 60 and 90 in Fig. S11) enables us to determine the mechanical phase φmec (shown in Fig. 4c for 1Hz). The small values of the mechanical loss tangent (0.02≤tan(φmec)≤ 0.06, see Fig. 4d) confirm the more elastic than viscoelastic response of La2Mo2O9 ceramics. The mechanical phase does not exhibit any peak or drastic change, neither around 10Hz (where the M coefficient switches sign) nor around 3Hz (where the strain changes sign). This underlines the fact that the mechanical properties are not responsible for either of the sign changes.

Therefore, it is the phase change of the electrostrictive response that causes the sign change of the electrostrictive M coefficient. If the experimental electrostrictive phase changes with frequency (as shown in Fig. 5a), the calculated electrostrictive phase change φx varies as a function of the damping parameter c. The evolution of the calculated φx with the damping coefficient c (see Fig. S7b) is analogous to the evolution of the experimental φx with frequency. From the combined evolutions of the experimental φx with frequency and of the calculated φx with c, it is possible to plot the frequency evolution of the damping coefficient. Figure 5b reveals that the damping coefficient evolves linearly with the excitation frequency. Such a linear evolution is consistent, for example, with a velocity-squared damping corresponding to a viscous damping proportional to the driving frequency (see Supplementary information S7). As a consequence, a frequency-dependent damping mechanism leads to a frequency-dependent phase variation. Figure 5 infers that “giant” electrostriction is not strongly dependent on polarization or elasticity. As both dielectric and mechanical properties show a continuous, stable evolution over the investigated frequency range, the variations of the electrostrictive coefficients are intrinsic in the sense that the electrostrictive coupling itself is frequency-dependent, through the frequency dependence of the damping coefficient.

Origin of the amplitude relaxation with frequency
We now turn to the frequency-dependent amplitudes to complement our description of the frequency dependence of the (complex) electrostrictive coefficients.

The amplitude of the electrostrictive coefficient in “giant” electrostrictors has so far been described as a non-ideal Debye relaxation (e.g., in oxygen-deficient ceria10,13 or acceptor-doped barium zirconate (BaZrO3)37): the amplitude of M exhibits a plateau at low frequency before monotonically decreasing with increasing frequency.

The amplitude of LAMOX electrostrictive coefficients is qualitatively different from the non-ideal Debye relaxation, varying little at low frequency before exhibiting a peak (even in absolute values, see the empty symbols in Fig. 6) and then a decrease with increasing frequency. The polarization electrostrictive coefficient, Q, exhibits the same behavior, see Supplementary information S8. From the anharmonic oscillator pair model, the amplitudes of the deformation are determined by three physical factors: the frequency-independent ratio of a/k3 (see Supplementary information S2), the damping coefficient c, and the ratio of the excitation frequency over the natural frequency (λ=ω/ω0). As the excitation frequency increases, the increase in the damping coefficient eventually decreases the strain amplitude (see Fig. S7d). When the excitation frequency approaches the natural frequency, the amplitudes can present a peak or relaxation. However, the numerical solutions of nonlinear oscillator models like the one used above can be unstable between the so-called jump-up and jump-down frequencies, which cannot occur in practice38,39. Analyzing the frequency-amplitude relationship near resonances in a nonlinear system where the damping coefficient changes with frequency is therefore beyond the reach of our model.

Nevertheless, the frequency-dependence of the electrostrictive coefficient amplitude can be rationalized by studying the amplitude of the resonance in a linear system. The variation of the amplitude A with the reduced frequency λ is then given by:

A(λ)∝ 1/√(λ4+(4ζ2−2)λ2+1) −α (6)

where ζ=c/2√mk is the damping ratio and α is the non-ideal resonance coefficient determining the width of the amplitude peak (the influence of ζ and α on the amplitude is illustrated in Supplementary information S9). Using Eq. (6), we could fit the frequency evolution of the electrostrictive coefficient. The best fit for M is shown with a dashed line in Fig. 6, where the value of α (α=0.14) underlines the non-ideality of the damped oscillator and ζ=0.18<1/√2 is consistent with the existence of a peak at the resonance frequency (ωres =ω0√1−2ζ2). On the contrary, if the damping ratio had exceeded 1/√2, the amplitude would have monotonically decreased, similar to the “Debye-type relaxation pattern” observed so far in other “giant” electrostrictors.

Therefore, in “giant” electrostrictors, the frequency-evolution of the amplitude is also imposed by the damping. Since the damping ratio is inversely proportional to the square root of the effective dipole mass (ζ=c/2√mk), the dipoles in La2Mo2O9 are heavier than the ones in Gd-doped ceria, positing that the stiffness k between oxygen-vacancies and rare-earth ions is similar in La2Mo2O9 and in ceria-based electrostrictors. Such a conclusion is consistent with the fact that only Nd-, Sm-, Gd-, and Er-doped ceria exhibit a Debye-type relaxation pattern, while the electrostrictive coefficients of heavier Yb- and Lu-doped ceria start increasing after 50Hz10. With our model, a peak is expected to occur in Yb- and Lu-doped ceria at higher frequencies. Moreover, the much broader frequency response of Zr-doped ceria (from 0.1Hz to 1kHz)11 may be related to the low mass of Zirconium.

Methods
Sample preparation
LAMOX ceramics were prepared from powders synthesized by solid-state reaction. Initial powders of stoichiometric amounts of elementary oxides (La2O3, MoO3) were ground into ethanol in an agate mortar, to obtain a homogenous mixture of oxides. The thermal treatment was carried out in an alumina crucible, first at 500°C for 12h to remove the organic binder and then at 900°C for sintering 24h. Pellets were shaped by uniaxial pre-pressing, followed by isostatic pressing at ~3500 bars. The optimized sintering temperature is 900°C for 2h. The phase purity of the samples was verified by X-ray diffraction and Raman Spectroscopy from our previous report36.

Electromechanical measurements
Electromechanical measurements were carried out on an AixAcct FerroTester TF2000E. The polarizations were measured using a PUND electric field sinusoidal profile and confirmed by reversing the polarity of the field as well as with a triangular profile. The dielectric permittivity values were extracted from the linear responses of the average slope of polarization under 5kV/mm. The strains were calculated from the surface deformation measurements and the thickness of the sample and plotted considering the first four harmonics. The phases were extracted from the time evolution of polarization or strain induced by a sinusoidal field signal.

Nano-indentation
Nanoindentation measurements were carried out with a Berkovich indenter from Anton Parr NHT3. The mechanical load pattern is divided into four sections: in the first section, the load increases linearly up to 50mN in 30s. Likewise, the unload (fourth) section decreases linearly with the same unloading rate, determining Young’s modulus from the standard Oliver-Pharr42 method. The differences in measured values between “fast” mode (5mN/s) and “slow” mode (0.15mN/s) in Pr-, Gd-, Lu-doped ceria43 and Sm-doped ceria44 have been reported to be less than 20%, i.e. within the range of the error bar. The “medium” loading rate of 1.67mN/s in this experiment is chosen since the loading rate does not strongly affect the measurement of Young’s modulus. The measurements of the dynamic elastic modulus and losses were carried out under a sinusoidal mechanical load of 2.5mN amplitude from 1Hz to 40Hz (see Supplementary information S11).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This work reported a new finding that the electrostrictive coefficients must be treated as complex values with their sign dictated by their phases rather than by the overall shape of the strain versus electric field curve for the non-classical electrostrictors. I think this finding is interesting. However, the structure origin of this finding is not discussed clearly, and some other key issues also need to be addressed clearly before consideration for publication.\n1. Keywords, giant electrostrictors? It should be “giant electrostrictive coefficients”? Besides, it is difficult to find the results obtained in this work from current keywords.\n2. What is the intrinsic meaning of “phase” in this work? Can the phase change induce a change from tensile strain to compressive strain? How do we change the phase?\n3. From Figs. 1 and 2, why is a huge strain hysteresis shown from π/8 to π3/8? Does it just result from a mathematical reason or some structural origin? Importantly, it is difficult and unscientific to discuss the electrostrictive behavior if the obvious strain hysteresis is shown.\n4. The classical ferroelectric electrostrictors also seem to show phase- or frequency-dependent electrostrictive strain and electrostrictive coefficients from SI. The structure origin should be the delay of domain switching along the applied electric field? Importantly, it is difficult and unscientific to discuss the electrostrictive behavior if the domain contribution is significant on strain. In addition, it seems that no similar phenomena have been found in many previous reports about similar classical electrostrictors, which also show the strain or electrostrictive properties under different frequencies.\n5. Why are such huge electrostrictive coefficients shown in La2Mo2O9 compared to classical dielectric electrostrictors? What is the structure origin of the huge electrostrictive coefficients in La2Mo2O9? Do their special structure origin contribute to their obvious phase- or frequency-dependent electrostrictive behaviors? What is the structure origin of the phase- or frequency-dependent electrostrictive behaviors?\n6. The article format must meet the requirements of this journal, especially for the reference format.\n7. There are some typos in the current manuscript. The authors should check them and revise the manuscript carefully.", "reviewer_2": "Interesting paper on electrostriction. I have the following points.\nElectric field - could be in kV/mm or kV/cm rather than MV/m, but this is just a personal prefernece.\nMore detail on Fig1 to introduce to the reader - on how this data is generated and the case conditions/assumption - is it derived from the equations or experimental?. The text in section 2 describes specifics e.g. PMN-10PT and oxygen-deficiant certia and then outlines a generic case via Fig 1. This could be much clearer to the reader. e.g. consider an electric field across a general material for different phases of the electric response, which may arise from various loss mechanisms such as…\nPaper would benefit from a clear definition of loss angle as it underpins most arguments in the paper (inc dielectric/mechanical/electrp-mechanical)\nLink between sections 2 (M sign and strain decoupled) and 3 (giant electrostrictors) could be strengthened but I see there is some attempt.\n“As expected, the phase of the electrostrictive response changes under decreasing excitation frequencies.” - help the reader.\n“As expected, the phase of the electrostrictive response changes under decreasing excitation frequencies since…..”\n[explain, as this will help reader].\nIt says “the strain switches from tensile to compressive between 2 and 3 Hz.” - where? what figure?\n“Probably less expected is that the sign of the electrostrictive coefficient also changes, albeit at another frequency (between 9 and 10 Hz)” - what figure, figure 2? Is this experimental data?\nSymbols re small fig 2 - not easy to read the subscripts.\n“… M coefficient could be related to a corresponding change in the mechanical properties” - such as? examples? Please add some text to help reader.\n“Nevertheless, the sign of the electrostrictive coefficient can be determined from the\nrotation sense of the hysteresis, it is known” - sentence not clear - please revise.\nDiscussion has a focus on what is not the reason ( e.g. Maxwell/field amplitude, but does not mention mechanical properties either)- but could be more clear what ARE the potential reasons.\nDisucssion or outlook can also advise what researchers should do or take care with, when measuring the electrostrictive\ncoefficients of their to understand their materials.\nInteresting work, could be made easier to read/understand.\nChris Bowen", "reviewer_3": "The manuscript reveals that the electrostriction is just a phase, offering a novel and significant contribution to electrostriction research. The results are well-supported by experimental data, and the authors present their findings clearly and concisely. However, revisions are required for publication in Nature Communications. My concerns are as follows:\n1. The authors should provide a more comprehensive explanation of the physical mechanisms underlying the phase-dependent behavior of electrostrictive coefficients. This would help to strengthen the argument that the observed phenomena are intrinsic to the material.\n2. The model introduced to explain the frequency-response of electrostrictive coefficients is an interesting addition, but the manuscript lacks detailed validation of this model. The authors should provide more information on how the model was derived and how it compares to experimental data, such as attempting to fit the experimental data using the model.\n3. The electromechanical measurements were carried out on an AixAcct FerroTester TF2000E, and the strains were calculated from the surface displacement measurements and the thickness of the sample. The authors should also discuss response times in terms of both the sample and the instrument.\n4. In sample preparation section, the thermal treatment was carried out in an alumina crucible, first at 500 ℃ for 12 h and then at 900 ℃ for 24 h. The reason for the initial 500°C for 12 h heat treatment should be explained. Additionally, XRD and Raman Spectroscopy data should be included."}, "author_response": "To the editor and three reviewers: thank you for pointing out the concern about the physical origin of the observed behavior of the electrostriction coefficients.\nPhysical background and origin:\nNon-classical (“giant”) electrostrictors have been reported, such as doped ceria (Gd, Zr, Sm), Y-stabilized bismuth oxide, La Mo O and itssubstitutions, La Ce O , and defective non-ferroelectric 2 2 9 2 2 7\nepitaxial BaTiO The existence of oxygen vacancies is a necessary and not sufficient condition 3.\nfor “giant” electrostriction. For instance, Y-stabilized ZrO ceramics do not exhibit “giant” 2\nelectrostriction in Ref.4 despite the presence of oxygen vacancies. It is not specific to fluorite-type structures. We have recently investigated perovskites with site occupation deficiencies and none exhibited a “giant” electrostrictive response (article under review).\nOne of the popular explanations of “giant” electrostriction is that the oxygen vacancies distort the crystal, modifying either from the hopping behavior (Ref.15) or the redistribution of mobile oxygen vacancies (Ref.4). However, these explanations remain qualitative. To respond to your comments, our article now includes a model for the electrostrictive response. It contains a forced, damped, anharmonic oscillator model that explains the phase- or frequency-dependent “giant” electrostriction from the motion of oxygen vacancy dipoles (or any other active dipole responsible for electrostriction) in an anharmonic potential well.\nRoadmap:\n1. Fig.1 shows the 1D anharmonic oscillator dipole pairs to present the origin of electrostriction. The schematic of the model can explain all the characteristics of electrostriction: (1) the direction of deformation is independent of the polarity of the electric field, the displacement responses double the frequency of excitation, (2) Electrostriction exists in both centrosymmetric and non-centrosymmetric systems, (3) Electrostriction has no reverse effect.\n2. Fig.2 demonstrates the numerical results of the net displacement of two opposite dipoles (indicating the mirror symmetry by 𝛿(𝑡) = 𝛿 (𝑡)+𝛿 (𝑡)). Fig.2a exhibits the transient and steady- 𝑟 𝑙\nstate responses of the Helmholtz oscillator with various damping coefficients c. Fig.2b and 2c illustrate the 𝛿(t) curves and 𝛿-E curves after the baseline correction of the setup (When t=0, 𝛿 should be 0). The values and units of the parameters are selected based on the experimental results of LAMOX. The details are in Supplementary Information S2. The goal here is to show that different values of the damping coefficient c will create the phase-dependent electrostrictive responses from the numerical simulation. And this is the reason for the sign change. In addition, the validity of this model is verified by experimental values and the interatomic potential of alkali halide in Supplementary Information S3.\n3. Fig.3 illustrates the experimental measurements of LAMOX with x-E curves and x as a function of time to stress the sign reversal experimentally within a short range of frequencies. We intensify the connections between experimental results and numerical simulations in the manuscript.\n4. Fig.4 excludes the influences of the dielectric (permittivity) and mechanical (elasticity) properties with slight variation of both real and imaginary parts.\n5. Fig.5 shows that only electrostrictive phases vary drastically, indicating “giant” electrostrictors are not strongly dependent on polarization or elasticity. The sign change phenomenon results from the linear frequency-dependent damping coefficients (Supplementary Information S7).\n6. Fig.6 presents the frequency-dependent amplitudes of electrostriction, which are also related to the damping coefficients (Supplementary Information S9).\nNovelties:\n1. First report of sign reversal behavior via frequencies and a possible explanation of the inconsistency of the sign of the electrostrictive coefficient reported in different references for the same materials.\n2. A more precise model of electrostriction combined with nonlinear dynamics.\n3. Velocity-squared damping (frequency-dependent damping) nature from the oxygen-vacancy-induced dipoles of “giant” electrostrictors.\nReviewer #1: This work reported a new finding that the electrostrictive coefficients must be treated as complex values with their sign dictated by their phases rather than by the overall shape of the strain versus electric field curve for the non-classical electrostrictors. I think this finding is interesting. However, the structure origin of this finding is not discussed clearly, and some other key issues also need to be addressed clearly before consideration for publication.\nWe thank this reviewer for the time they dedicated to reading this manuscript.\n1. Keywords, giant electrostrictors? It should be “giant electrostrictive coefficients”? Besides, it is difficult to find the results obtained in this work from current keywords.\nA: We thank the reviewer for this suggestion, which will make our work easier to find from a keyword search. We have changed the keyword according to the suggestion.\n2. What is the intrinsic meaning of “phase” in this work? Can the phase change induce a change from tensile strain to compressive strain? How do we change the phase?\nA:\nWe adopt the mathematical definition of the phases from Eq.2 and Fig.4a (the phase of polarization 𝜑 ), Eq.3 and Fig.3b (the phase of electrostriction 𝜑 ), Eq.5 and Fig. 4c (the phase 𝑒𝑙𝑒𝑐 𝑥\nof elasticity 𝜑 ). Specifically, we refer to the time difference between the response and the 𝑚𝑒𝑐\nexcitation (for electrostriction, between the strain and the electric field, for the susceptibility between the polarization and the electric field, etc.). To ensure causality, these phases must be positive.\nThe phase change is indeed the origin of the sign change of the electrostrictive response. Since the phase is measured with respect to the electric signal, and that this phase must be positive, as the strain curve “slides” to the right, a maximum electric field will correspond to either a maximum strain (leading to a positive electrostrictive coefficient) or to a minimum strain (leading to a negative coefficient).\nWe show that the frequency evolution of the phase (Fig.5a) corresponds to the frequency dependence of the damping coefficient (Fig.S7b), indicating that the damping coefficient value varies linearly with frequency (Fig.5a). So, to change the phase in LAMOX, the excitation frequency should be changed to change the damping coefficient.\n3. From Figs. 1 and 2, why is a huge strain hysteresis shown from π/8 to π3/8? Does it just result from a mathematical reason or some structural origin? Importantly, it is difficult and unscientific to discuss the electrostrictive behavior if the obvious strain hysteresis is shown.\nA:\nWe thank the reviewer for pointing out that our presentation was confusing. We added a physical model to explain that if the damping coefficients can play an important role of the strain hysteresis. For example in Fig.2b, the π/8 corresponds to c=3, π3/8 corresponds to c=12.\nFig.2 is the mathematical calculation and Fig.3 is the experimental strain response of La Mo O 2 2 9\nfor different frequencies. In Fig.3, with the decrease of the frequency, the La Mo O sample 2 2 9\nexhibits electrostrictive strain changing from extension to compression. With a larger damping coefficient at high frequency, the sample compresses less as the ac field increases and compresses more as the ac field decreases.\nIn the measurement setup (for us, an interferometer), strain is always considered zero at the initial time, causing the strain-field curve to be an opening upward parabola, corresponding to tensile responses. When the frequency drops to around 1Hz, the electrostrictive displacement is mainly compressive.\nThe evolution of x-E curve from high frequency to low frequency corresponds to the steady-state solution of the anharmonic oscillator pair model described at the beginning, when the damping coefficient c decreases as the driving frequency decreases.\nTo conclude the reviewer’s question: a huge strain hysteresis is indeed the measured response of the sample, as shown in the manuscript in Fig.3. To explain the sign change phenomenon, we brought in numerical solutions of our model in Fig.2 to explain that the frequency variation of the damping coefficient induces such behavior.\n4. The classical ferroelectric electrostrictors also seem to show phase- or frequency-dependent electrostrictive strain and electrostrictive coefficients from SI. The structure origin should be the delay of domain switching along the applied electric field? Importantly, it is difficult and unscientific to discuss the electrostrictive behavior if the domain contribution is significant on strain. In addition, it seems that no similar phenomena have been found in many previous reports about similar classical electrostrictors, which also show the strain or electrostrictive properties under different frequencies.\nA:\nWe thank the reviewer for mentioning the comparison of classical ferroelectric electrostrictors. The reviewer is right, the structural origin in classical ferroelectric electrostrictors is most probably, the delay of domain switching, resulting in the variation of polarization and electrostrictive response.\nDomains have a significant contribution to strain, and therefore, it is not an appropriate system to investigate electrostriction as it is piezoelectric. The responses of PMN-10PT are primarily electrostrictive (x-t and x-E curves in Fig.S5a and b) to discard piezoelectricity. Indeed, there is a frequency dependence of the permittivity related to domain switching, and this phenomenon would contribute to the overall response, but it is difficult to disentangle the two.\nThe goal of Supplementary Information S5 about PMN-10PT is to show the consistency of the redefinition of the sign of electrostrictive coefficients. Even though the physical origins of electrostrictive responses in PMN-10PT (polar-nano domains or polar-nano regions, etc.) and “giant” electrostrictors (oxygen-vacancy-induced dipoles) are different.\nThe reviewer is right, we have not come across similar reports on other electrostrictors. Actually, our reports show that the frequency dependence of ceria and LAMOX can be described using the same model. Specific behaviors are related to the underlying interatomic potential and the specific values of its parameters.\n5. Why are such huge electrostrictive coefficients shown in La2Mo2O9 compared to classical dielectric electrostrictors? What is the structure origin of the huge electrostrictive coefficients in La2Mo2O9? Do their special structure origin contribute to their obvious phase- or frequency-dependent electrostrictive behaviors? What is the structure origin of the phase- or frequency-dependent electrostrictive behaviors?\nA:\nWe thank the reviewer for mentioning the question of why oxygen-deficient materials exhibit large electrostriction (10-4 of strain in ceramics and 10-2 of strain in thin films). The answer currently put forward is that oxygen vacancies create local distortion under an electric field, and the amplitude and phase of the strain are strongly related to the driving frequency. For oxygen-deficient thin film materials, Ref.4 reported that Gd-doped ceria thin films exhibiting large electrostriction originate from the mobile and polarized oxygen vacancies due to the reduction of the thickness and strong bias or ac field.\nHowever, not all oxygen-vacancy-rich materials exhibit this “giant” electrostriction. For example, Y-stabilized ZrO (8% or 10%), which has a pseudo-cubic fluorite structure. “Giant” electrostrictors 2\nwith certain crystal structures and oxygen vacancies include pseudo-cubic fluorite (Doped ceria, Stabilized Bi O ), SnWO -like cubic and monoclinic structure (La Mo O family), perovskite 2 3 4 2 2 9\n(defective non-ferroelectric epitaxial BaTiO , cation-deficient BaZrO ). 3 3\nThe structure will definitely influence the mass, spring constant, anharmonicity, and damping of the electrical dipoles induced by oxygen vacancies, which in turn affect the phase- or frequency-dependent electrostrictive behaviors.\n6. The article format must meet the requirements of this journal, especially for the reference format.\nA: We thank the reviewer for mentioning the format. We have changed them in order to meet the requirements of the journals.\n7. There are some typos in the current manuscript. The authors should check them and revise the manuscript carefully.\nA: We thank the reviewer for mentioning the typos. We had a major revision to the last version of the manuscript and have corrected, with the help of a native speaker.\nReviewer #2 (Remarks to the Author):\nInteresting paper on electrostriction. I have the following points.\nWe thank the reviewer for showing interest and carefully reading this manuscript, for the opportunity to improve our paper through your perspective as a reader, and to improve the presentation of the ideas and the logic of the paper.\nElectric field - could be in kV/mm or kV/cm rather than MV/m, but this is just a personal preference.\nA: We thank the reviewer for your suggestion. Indeed, kV/mm is more straightforward as a unit for the electric field when dealing with ceramics. We have changed all the units of the electric field into kV/mm.\nMore detail on Fig1 to introduce to the reader - on how this data is generated and the case conditions/assumption - is it derived from the equations or experimental?. The text in section 2 describes specifics e.g. PMN-10PT and oxygen-deficiant certia and then outlines a generic case via Fig 1. This could be much clearer to the reader. e.g. consider an electric field across a general material for different phases of the electric response, which may arise from various loss mechanisms such as…\nA:\nWe thank the reviewer for mentioning this confusion. We have totally revised the old Fig.1. The revised manuscript contains Fig.1 for the schematic of the anharmonic oscillator pairs and Fig.2 for numerical simulations with different damping coefficient values. There are three columns in this mathematical and physical model:\n(1) The left column (Fig.2a) is the displacement 𝑥(𝑡) and the electric field 𝐸(𝑡) as a function of time by solving of differential equation when the viscous damping coefficient c varies. Every parameter is given the values and units in the atomic level, described in Supplementary Information S2.\n𝑘 𝑎 𝑐 𝑞𝐸\n𝑥̈ + 𝑥+ 𝑥2+ 𝑥̇ = ± 0 sin(𝜔𝑡) → 𝑥(𝑡) = 𝑥 (𝑡)+𝑥 (𝑡)\n𝑚 𝑚 𝑚 𝑚 𝑟 𝑙\n(2) The middle column (Fig.2b) is the corresponding 𝑥(𝑡) curves in the steady-state region (different damping coefficients c), shifting 𝑥(𝑡) to 𝑥(𝑡 = 0) = 0 , which is the expected electrostrictive response with the interferometer setup (strain is always considered zero at the initial time as shown in the equation).\n𝑥(𝑡) = 𝑀𝐸2sin2(𝜔𝑡−𝜑 )−𝑀𝐸2sin2𝜑\n0 𝑥 0 𝑥\n(3) The right column (Fig.2c) is the shifted displacement 𝑥 as a function of 𝐸\nWe also added the descriptions of the frequency-dependent damping coefficient in Fig.5 and Supplementary Information S7, which is the reason for the sign change.\nPaper would benefit from a clear definition of loss angle as it underpins most arguments in the paper (inc dielectric/mechanical/electro-mechanical)\nA: We add the mathematical definition of the phases from Eq.2 and Fig.4a (the phase of polarization 𝜑 ), Eq.3 and Fig.3b (the phase of electrostriction 𝜑 ), Eq.5 and Fig. 4c (the phase 𝑒𝑙𝑒𝑐 𝑥\nof elasticity 𝜑 )\n𝑚𝑒𝑐\nLink between sections 2 (M sign and strain decoupled) and 3 (giant electrostrictors) could be strengthened but I see there is some attempt.\nA: We thank the reviewer for pointing this out. Our intention is indeed to present the sign change behavior (in Section 3) using a mathematical and physical model to explain it (in Section 2). Hence, in the introduction part, we have added a description of our intention at the end of the introduction. We have revised the content of Sections 2 and 3 to have a (hopefully) clearer correlation and comparison. For example, by listing the values of c=0.05, 3, 6.7, etc in numerical simulation, we find the correspondences in the experimental measurements in Fig.3a.\n“As expected, the phase of the electrostrictive response changes under decreasing excitation frequencies.” - help the reader. “As expected, the phase of the electrostrictive response changes under decreasing excitation frequencies since…..” [explain, as this will help reader].\nA: Thank you again for helping us make our article clearer. We have deleted the sentence before and added the sentence: “Here, we demonstrate experimentally the frequency-dependent sign change of the overall shape of the x-E curves on bulk (ceramics) of pure La2Mo2O9 “giant” electrostrictor.”\nThe sentence here is the transitional sentence of frequency-dependent sign change from the simulation results to the experimental results.\nIt says “the strain switches from tensile to compressive between 2 and 3 Hz.” - where? what figure?\n“Probably less expected is that the sign of the electrostrictive coefficient also changes, albeit at another frequency (between 9 and 10 Hz)” - what figure, figure 2? Is this experimental data?\nA: We apologize for these missing references to the corresponding Figures. They have been added in the new version.\nWe describe that “mainly” tensile and compressive electrostrictive strain is the opening direction of the parabola. The concave upward of the x-E parabola in Fig.2c indicates the tensile responses, and the concave downward of the x-E parabola in Fig.2c indicates the compressive responses.\nThe experimental results in Fig.3 show that the transition from tensile (concave upward) to compressive (concave downward) is indeed between 2Hz and 3Hz.\nAkin to the previous definition, we redefine the sign of M coefficients by analyzing the correspondences of E(t) and x(t).\n“The sign of the M coefficient is defined as positive when the maximum of the strain corresponds to the maximum of the electric field. This correspondence holds as long as the maximum strain occurs after the maximum of the electric field and before the next minimum electric field. Inversely, the M coefficient is, negative when the maximum of the strain occurs after a trough of the field and before the next maximum.”\nThe experimental results in Fig.3 show that the transition from M>0 (the peak of E corresponds to the peak of x) to M<0 (the peak of E corresponds to the trough of x) is between 9Hz and 10Hz.\nThese two critical points simulated in Fig.2 (c=6.7 and c=22) correspond to the experimental results (3Hz and 10Hz).\nSymbols are small fig 2 - not easy to read the subscripts.\nA: We have changed Fig.2 (now Fig.3) and increased the font to make the subscripts more legible\n“… M coefficient could be related to a corresponding change in the mechanical properties” - such as? examples? Please add some text to help reader.\nA: Thank you for pointing out the lack of precision. The sentence now reads:\n“Since electrostriction is an electromechanical coupling, the sign change of the strain and/or the M coefficient could be a consequence of frequency-dependent changes in the dielectric or mechanical properties.”\n“Nevertheless, the sign of the electrostrictive coefficient can be determined from the rotation sense of the hysteresis, it is known” - sentence not clear - please revise.\nA:\nThank you for pointing out the lack of precision. Here we wanted to present that only with x-E curves, the physical parameters are hidden. For example, in Fig.2c the case of c=12 and c=48, x-E curves are exactly the same. However, by indicating the rotation direction of the x-E hysteresis, we can identify the case of M>0 or M<0.\nIn the manuscript, we deleted this description, and we added the following sentence to convey this point, now read:\n“Note that identical x-E curves can be obtained for various values of a and c (see Fig.S4c). This suggests that the physical significance of electrostriction may be obscured if only the x-E curves are considered.”\nThe discussion has a focus on what is not the reason ( e.g., Maxwell/field amplitude, but does not mention mechanical properties either)- but could be more clear what ARE the potential reasons.\nA:\nThe discussion about what is not the reason is equally important because we want to make sure the measured quadratic strain does not stem from the electrostatic attraction of the setup or the Maxwell strain of the samples (the electrostatic attraction from the top and bottom electrodes).\nWe hope the model presented at the beginning answers this comment about what the potential reasons are.\nDiscussion or outlook can also advise what researchers should do or take care of when measuring the electrostrictive coefficients of their to understand their materials.\nWe have added a paragraph on section outlook (The first two paragraphs) to provide such advice:\n“To unambiguously determine the sign of electrostrictive coefficients from the induced deformation under large electric fields, it is essential to simultaneously acquire the time evolutions of both the electric field and the displacements. This approach will clarify whether the strain extremum, following the field maximum, is also a maximum (indicating a positive electrostrictive coefficient) or a minimum (indicating a negative electrostrictive coefficient). Then, taking into account the possible contribution from the Maxwell stress, the field dependence of the electrostrictive strain can be plotted. To extract the amplitude of the electrostrictive coefficient, the potential contributions from higher-order terms need to be considered (see Ref.[17]) and removed if necessary to achieve the expected linear dependence of the electrostrictive (i.e., second-order only) strain on the square of the electric field. When comparing measured electrostrictive coefficient values with those calculated ab initio[24], it is important to make sure the so-called “proper” electrostriction coefficients have been calculated[40].\nTo analyze the frequency dependence of the electrostrictive response, it is necessary to measure the frequency-dependent dielectric and mechanical properties in addition to the frequency-dependent displacement. This will help separate the purely dielectric or mechanical contributions from the purely electrostrictive ones.”\nInteresting work, could be made easier to read/understand.\nThank you. We hope this revised version is easier to read and understand.\nReviewer #3 (Remarks to the Author):\nThe manuscript reveals that the electrostriction is just a phase, offering a novel and significant contribution to electrostriction research. The results are well-supported by experimental data, and the authors present their findings clearly and concisely. However, revisions are required for publication in Nature Communications. My concerns are as follows:\n1. The authors should provide a more comprehensive explanation of the physical mechanisms underlying the phase-dependent behavior of electrostrictive coefficients. This would help to strengthen the argument that the observed phenomena are intrinsic to the material.\nWe thank the reviewer for pushing us to come up with the model presented at the beginning of this document.\n2. The model introduced to explain the frequency-response of electrostrictive coefficients is an interesting addition, but the manuscript lacks detailed validation of this model. The authors should provide more information on how the model was derived and how it compares to experimental data, such as attempting to fit the experimental data using the model.\nWe share the reviewer’s concern about experimental validations of models. We have verified this anharmonic oscillator pair model with the experimental results of alkali halides in Supplementary Information S2 and S3.\nThe frequency-dependence of the M electrostrictive coefficient has indeed been fitted to the experimental results and presented as a dotted line in Fig.6.\n3. The electromechanical measurements were carried out on an AixAcct FerroTester TF2000E, and the strains were calculated from the surface displacement measurements and the thickness of the sample. The authors should also discuss response times in terms of both the sample and the instrument.\nThis is a very insightful comment, and we appreciate it. The response time of the setup has been investigated under various excitation types and parameters (https://theses.fr/2022UPAST089) and can indeed influence results if overlooked. In addition, as shown in Fig.2a, the transitory response is considered in our model, which exists for no more than two periods. In addition, our measurements have been carried out by averaging over 10 cycles. All together, they ensure that the reported curves correspond to the steady state response of our model.\n4. In sample preparation section, the thermal treatment was carried out in an alumina crucible, first at 500 ℃ for 12 h and then at 900 ℃ for 24 h. The reason for the initial 500°C for 12 h heat treatment should be explained. Additionally, XRD and Raman Spectroscopy data should be included.\nThe first heat treatment is done to avoid molybdenum oxide sublimation. It is not added to the manuscript. XRD and Raman spectroscopy data have been published in the meantime (https://doi.org/10.1002/adfm.202500821) and the reference has been added in the experimental part."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors respond to the reviewers' comments in detail and carefully revise the manuscript. I recommend accepting this paper in its current form.", "reviewer_2": "Good overall response to referee comments and the paper is improved and is more clear, this is in terms of the discussion, conclusion and the figures. Interesting work.", "reviewer_3": "The author has provided satisfactory responses to all queries, and the paper is deemed acceptable."}, "author_response": "We thank the reviewer for the approval of this work.\nWe thank the reviewer for the interest of this work.\nWe thank the reviewer for the effort and response of this work."}]

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