Diffusional dynamic process of irreversible ferroelectric to antiferroelectric phase transition and the rate limit

Domain: physics
Published: 2025-12-31
Editorial outcome: minor_revision

Abstract
The ferroelectric to antiferroelectric phase transition in perovskite structures and its underlying mechanism form the physical foundation for numerous applications such as high-performance energy storage, ultrafast information storage, and electrocaloric refrigeration. Understanding of ferroelectric to antiferroelectric phase transition, especially the description of the transition dynamic process, is crucial for material design and device reliability enhancement. Most of the previous studies demonstrated that ferroelectric to antiferroelectric phase transition are first-order transitions driven by atomic rearrangement, generally regarded as prototypical fast kinetic processes. Here, we report an irreversible ferroelectric to antiferroelectric phase transition at low temperatures in Pb(Zr0.97Ti0.03)O3 as seen by neutron diffraction patterns. The in-situ variable-temperature transmission electron microscopy reveals that the ferroelectric to antiferroelectric phase transition exhibits relaxation behavior and is a slow diffusional dynamic process, which corrects the previous understanding. Density functional theory simulations reveal that the irreversible transition behavior is due to variation of the transition energy barrier with temperature. Based on Density functional theory and experimental data, a multiscale phase-field model was developed to describe the transition dynamics and provide the theory limitation for the rate of ferroelectric to antiferroelectric phase transition in Pb(Zr0.97Ti0.03)O3 at low temperatures, offering a physical basis for future material applications and device design.

Introduction
The ferroelectric to antiferroelectric (FE-AFE) phase transition involves crystal symmetry changing, dipole reordering, and significant changes in energy and entropy, representing a frontier interdisciplinary field in condensed matter physics and materials science1–5. The dynamic mechanism of the FE-AFE phase transition in perovskite structures forms the physical foundation for applications such as high-performance energy storage, ultrafast information storage, and electrocaloric refrigeration6–9. For instance, the giant isothermal entropy change (electrocaloric effect) generated during the FE-AFE transition enables all-solid-state refrigeration10–12; the instantaneous release of substantial polarization energy accompanying the FE-AFE transition underpins the ultrahigh energy storage density in antiferroelectric capacitors13–15. Critically, these functionalities are inherently governed not only by the initial and final phases but also by the kinetics and pathways of the transition process itself. Therefore, a deep understanding of the FE-AFE phase transition — particularly a detailed depiction of its dynamic evolution is essential for guiding rational material design, optimizing functional performance, and ensuring long-term device reliability.

To date, there are two main approaches that reveal the structural evolution of the FE-AFE phase transition: static structural observation and analysis of dynamic evolution. Static structural observation focuses on characterizing the stable phases before and after the transition, predominantly utilizing techniques such as Transmission Electron Microscopy (TEM) and various diffraction methods16–23. For example, the AFE-FE transformation of (Bi0.5Na0.5)TiO3-0.07BaTiO3 was revealed by using Neutron Powder Diffraction (NPD) technology16. In Sm-doped BiFeO3 materials, High-Angle Annular Dark-Field Scanning Transmission Electron Microscopy (HAADF-STEM) was applied to statically observe the FE-AFE structure transformation under an electric field17. Although TEM techniques are effective for observing atomic-scale phase structures, dynamically capturing the phase transformation process is challenging, necessitating the waiting for opportune moments and requiring rapid response capabilities.

Dynamic evolution studies of FE-AFE phase transitions rely heavily on electrical characterization techniques and theoretical simulations to infer the kinetics, such as analyzing current transient response curves or performing computational modeling10,16,24–32. For instance, in PbZrO3-based ceramics, high-time-resolution strain measurements revealed that the field-induced AFE-FE transition is an almost instantaneously lattice adjustment process16. The FE-AFE phase transitions in BiFeO3 ceramics could be completed instantly by measuring the strain16. The field-induced AFE-FE phase transition of AgNbO3 was an instantaneous process shown by the electrical response24. Time-Dependent Density Functional Theory - Molecular Dynamics (TDDFT-MD) simulations indicated that a laser pulse can trigger polarization switching in CuInP2S6 within approximately 400 femtoseconds25. Using voltage pulse excitation and broadband current response measurements, it was observed that the AFE-FE phase transition in Hf0.25Zr0.75O2 (HZO) thin films can be completed in approximately 2-4.5 nanoseconds26. Most dynamic studies have demonstrated that FE-AFE phase transitions are first-order phase transitions driven by atomic rearrangements, generally regarded as prototypical fast kinetic processes. However, due to the lack of direct in-situ observational evidence, significant controversies remain in understanding the dynamic process of the FE-AFE transition, such as: What are the characteristics of phase evolution? How fast is the phase transition?

In this work, the neutron diffraction technology was applied to reveal that Pb(Zr0.97Ti0.03)O3-1%Nb2O5 (PZT97/3) would undergo an irreversible FE-AFE phase transition at about −20 °C, which updated the phase diagram of Pb(Zr1-xTix)O3 compositions (PZT). PZT ferroelectrics were chosen because they are one of the most famous perovskite oxides for their versatile applications and complex phase structures. Variable-temperature transmission electron microscopy (in-situ TEM) was employed to capture this FE-AFE transition process. The in-situ videos, for the first time, revealed that the FE-AFE transition exhibits relaxation behavior and is a diffusive dynamic process. Based on Density functional theory (DFT) simulations and experimental data, a multiscale phase-field model was developed to describe the transition dynamics and provide the theoretical limitation for the rate of FE-AFE transition in PZT97/3 at low temperatures, offering a physical basis for future material applications and device design.

Results
Through variable-temperature neutron diffraction measurements, we systematically investigated the crystal structural evolution of PZT97/3 during a cooling cycle from 25 oC to −50 oC and subsequent reheating back to 25 oC, as illustrated in Fig. 1a–c. At the initial state (25 oC), PZT97/3 exhibits the typical ferroelectric rhombohedral structure with space group R3c. Upon cooled to −50 oC, the ferroelectric structure transforms into an antiferroelectric phase with the space group Pbam. During the structural phase transition, new diffraction peaks emerge at approximately 2θ=48°, 56°, and 83°. These peaks are assigned to the 1/4(161)P, 1/4(-160)P, and 1/4(152)P planes, characteristic of the Pbam phase, providing clear evidence for the R3c-Pbam phase transition. Remarkably, this Pbam structure remains stable upon reheating back to 25 oC. Corresponding structural parameter changes are detailed in Table S1 and Fig. S1. Supporting X-ray diffraction patterns (XRD, Fig. S2) confirm the sequence of phase transitions R3c-Pbam-Pbam occurring during 25 oC → −50 oC → 25 oC. Furthermore, the dielectric temperature spectrum (Fig. S3) exhibits a sharp peak at approximately −20 oC, signifying the occurrence of the FE-AFE phase transition. This trend is closely related to lattice reconstruction and enhanced structural ordering. For piezoelectric ceramic materials, the conventional phase diagram indicates that during the cooling process from 300 K to 200 K, they will transform from the antiferroelectric phase to the ferroelectric phase (Fig. 1e). However, theoretical calculations and optical analyses by Li et al.33 suggest an additional FE-AFE phase transition occurring in the low-temperature region. In this work, we demonstrate that such a phase transition indeed exists by undoubted experimental evidence. This discovery has been experimentally verified to confirm that the update of the PZT phase diagram by Li et al. is correct. As shown in Fig. 1(e), we have redrawn the updated phase diagram. Figure 1f–h depict the unit cells change during the FE-AFE transformation, which is manifested as the displacement of Pb atoms along the a-axis direction, giving rise to the antiferroelectric phase. During the subsequent reheating process back to 25 oC, the atomic positions remain essentially unchanged, exhibiting a slight volumetric expansion of the unit cell (Table S1).

Building on the evidence of phase transition in PZT97/3, the in-situ TEM analysis provides unprecedented nanoscale insight into the structural evolution. At 25 oC, the selected area electron diffraction (SAED) pattern along the [110]p zone axis exhibits prominent 1/2{111} superlattice reflections34, confirmed the FE R3c phase (Fig. 2a, b). Upon immediately cooling to −50 oC, the R3c phase remains stable — evidenced by the SAED pattern showing exclusively sharp 1/2{111} reflections with no new features (Fig. 2c-d). Crucially, only after a subsequent 2-hour relaxation at −50 oC, the system could transit to an AFE Pbam phase, confirmed by the appearance of well-defined 1/2{001}, 1/2{110} and 1/2{111} superlattice reflections in the SAED pattern (Fig. 2e, f)35–39. The AFE state persists upon reheating to 100 oC, confirming that thermal cycling alone cannot reverse the FE-AFE transformation as shown in Fig. S4.

To probe the dynamics of the FE-AFE transition in real time, we collected time-resolved TEM images (Fig. 2g–l and Video S1). At t = 0 s, a sharp interface demarcates coexisting FE and AFE regions. As the transformation proceeds, AFE domains systematically advance into the FE matrix, with the propagation front locally obstructed at FE domain boundaries by t = 20 s (Fig. 2i), indicative of microstructural pinning. Despite this hindrance, this diffusion-controlled boundary migration fully consumes the FE phase within 50 s - a second-timescale transition revealing unexpectedly sluggish dynamics - and culminating in a homogeneous AFE state (Fig. 2l).

Atomically resolved high-angle annular dark field (HAADF)-STEM imaging (Fig. 3a, b) directly captures a phase boundary separating the rhombohedral FE (R3c) and orthorhombic AFE (Pbam) phases of PZT97/3, with their coexistence confirmed by wide-field atomic-resolution images and corresponding fast Fourier transform (FFT) patterns (Fig. S5). Remarkably, an atomically thin interfacial region exhibits structural signatures distinct from both bulk phases, suggesting a boundary phase (BP) that mediates the transition. Color-coded Pb–Pb distance maps (Fig. 3a) show a gradual modulation of lattice parameters across the interface: the BP exhibits only moderate, spatially heterogeneous distortions, in contrast to the severe distortions of the AFE Pbam region and the nearly undistorted FE R3c region. Simultaneously, atomic displacement vectors (Fig. 3b) trace the polarization evolution from parallel alignment in R3c to antiparallel alignment in Pbam. Within the BP region, periodic modulations resembling AFE order appear with uncompensated dipoles, indicating a coherent rotation of polarization with reduced amplitude. The scatter plot (Fig. S6), compiled from all vectors in Fig. 3b, reveals a continuous 180o rotation of vector orientations within the BP region, from ~45o to 225o, accompanied by a gradual decrease in magnitude, collectively signifying a suppressed yet coherent polarization transition.

As illustrated in Fig. 3c, the region undergoing the phase transition can be partitioned into three structurally distinct subregions, based on variations in lattice parameters and Pb displacements: R3c region (R), the boundary phase region (BP) and Pbam region (O). Structural reconfiguration occurs specifically within the BP region, which acts as a strained coherent interface. The BP creates a strain-buffering zone that hinders antiferroelectric domain growth across the BP, effectively acting as a dynamic barrier to the FE-AFE transition40. Consequently, the interfacial energy barrier fundamentally prohibits transformation pathways, constraining the system to a sustained diffusion-controlled process.

Line-profile analysis of lattice spacing (Fig. 3d, extracted from the pale yellow box in Fig. 3a) quantitatively illustrates the interface evolution. The R3c region shows an almost uniform spacing (Δd < 5 pm), whereas the BP exhibits pronounced fluctuations (~40 pm) before the profile transitions to large antipolar oscillations in the Pbam (~80 pm). This progression confirms that the BP functions as a structural-adaptor layer: it imposes intermediate local distortions to buffer the lattice mismatch and mediate symmetry reconstruction during the phase transformation. Pb displacement profiles provide further insight into polarization dynamics (Fig. 3e, extracted from pale blue box in Fig. 3b). In R3c, Pb ions exhibit uniform, large displacements (~30–35 pm) at a constant orientation, whereas in Pbam they alternate in direction by ~180o, reflecting antiparallel dipoles. Within the BP region, two distinct local displacement patterns emerge with a characteristic fourfold periodicity. In one pattern, pairs of Pb atoms retain large displacements similar to those in R3c, with local lattice distortions accommodating adjacent dipole reorientation. In the other, Pb displacements are strongly suppressed (~10 pm), akin to a paraelectric-like, frustrated polarization state.

Within the transitional region preceding the Pbam phase, polarization vectors gradually rotate, and Pb displacements evolve toward an antiparallel configuration, marking the dynamical onset of antiferroelectric order. In the region adjacent to the R3c phase, dipoles with reduced amplitude remain closely aligned with those in the untransformed R3c matrix, implying that polarization reversal occurs first by reducing displacement magnitude prior to directional rotation. Collectively, the Pb displacement data delineate a sequential transition pathway: the BP phase first suppresses dipole amplitudes, then facilitates their directional reversal. This stepwise coordination of lattice distortion and polarization rotation by the BP phase during the time-dependent evolution across the FE–AFE boundary mediates a diffusion-mediated phase transition, as evidenced by the progressive propagation of structural changes and polarization reorientations across the phase boundary. Critically, the lattice mismatch at the BP interface (quantified by 40 pm lattice fluctuations) establishes a substantial energy barrier. Overcoming this barrier necessitates the observed sequential suppression and rotation of dipoles, rendering the transformation a sustained diffusion-controlled process rather than a transient event.

To better understand this irreversible and relax phase transition process, the thermodynamic analysis has been carried out shown in Fig. 4. The calculated total energies of FE and AFE phase as a function of temperature through the method of phonon are displayed in Fig. 4a. It can be found that the rhombohedral R3c structure exhibits lower energy at high temperatures (> −10 oC), whereas the orthorhombic Pbam structure has lower energy at low temperatures (< −10 oC). This indicates that the FE phase is more stable at high temperatures, whereas the AFE phase is more stable at low temperatures. It is noteworthy that the energies of the two structures intersect around −10 oC, suggesting the potential for phase coexistence in this temperature range. Furthermore, the phase transition energy barriers between FE and AFE phases were calculated at different temperatures based on the BP structure captured in our experiments, as shown in Fig. 4b, c. It can be found that the energy barrier for the transition from FE to AFE is 25 meV at −50 oC. Consequently, this 25 meV energy barrier requires a relaxation time for the R3c structure to gradually transform into the Pbam structure. It is worth noting that the energy barrier ΔER-O decreases with decreasing temperature (Fig. S7). This suggests that the structural transition becomes relatively easier at lower temperatures. Consequently, as temperature decreases, the relaxation time shortens and the transition kinetics accelerate (Video S2). However, the energy barrier for the transition from AFE to FE is as high as 43 meV at 25 oC. Therefore, the temperature field effect alone is insufficient to overcome the 43 meV energy barrier and facilitate the reverse phase transition from Pbam to R3c. As show in Fig. S8, after the FE-AFE transition is completed upon cooling, reheating to room temperature does not trigger the reverse transformation under zero-field conditions, suggesting that thermal energy alone is insufficient to overcome the transition from AFE to FE at −50 oC. In contrast, when an external electric field is applied at room temperature, the AFE superlattice reflections disappear, indicating that the AFE-FE transition can occur under the applied field. This behavior is consistent with previous observations in PZT-based ceramics41–44. The intervention of other external field energy, such as an electric field or an irradiation field, is necessary for the AFE-FE transition.

To describe the dynamic process of the FE–AFE phase transition with high fidelity, a multiscale theoretical framework combining first-principles energetics and mesoscale kinetics was established. We conducted two-dimensional (2D) phase-field simulations to investigate the FE-AFE phase transition at low temperatures, with the free energy (ΔG) directly derived from density functional theory (DFT) calculations. In this framework, the DFT calculations provide the atomic-scale foundation for constructing the Landau-Ginzburg-Devonshire (LGD) free-energy functional. Specifically, the DFT-calculated total energies of the FE and AFE phases were fitted by a sixth-order Landau polynomial expansion, where the coefficients were optimized to reproduce the DFT-computed energy hierarchy and its temperature evolution. The temperature dependence of the second-order Landau coefficient follows the Curie-Weiss law, capturing the thermodynamic evolution of the free-energy difference between the FE and AFE phases. This DFT-based parameterization ensures that the phase-field simulations inherit a quantitatively accurate free-energy landscape derived from first principles, thereby bridging the atomic and mesoscale descriptions.

To ensure a physically consistent connection between atomic-scale energetics and mesoscale evolution, the parameterized Landau free-energy functions were incorporated into the phase-field simulations as the thermodynamic driving force for phase transformation, while the elastic, electrostrictive, and gradient energy coefficients were taken from experimental or literature values. The thermodynamic driving force arises from the local free-energy difference (Δf) between the FE and AFE states at a given temperature, which represents the functional derivative of the total free energy with respect to the polarization order parameter in the time-dependent Ginzburg-Landau (TDGL) equation. Physically, this quantity determines the direction and rate of domain evolution as the system seeks to minimize its total free energy.

Among these parameters, the gradient energy coefficient controls the domain-wall width and energy, where the kinetic coefficient (L) governs the dynamic response of domain-wall motion. In the TDGL framework, the diffusion of the domain wall is jointly determined by the thermodynamic driving force (Δf) and the kinetic coefficient L, modulated by the interfacial and elastic energy terms that affect the wall mobility and shape. Accordingly, the domain-wall velocity V follows a linear relationship with the local free-energy difference and the kinetic coefficient, expressed as V ∝ LΔG45,46, where L represents the temperature-dependent interfacial mobility.

To capture microstructural effects, localized perturbations were introduced into the order parameter field to mimic structural defects such as dislocations or compositional inhomogeneities. In the simulations, several grid points were randomly assigned small fluctuations (within ±5%) in the local polarization, thereby producing spatial variations in the free-energy landscape. These perturbations act as pinning centers or nucleation sites, leading to curved domain walls, interface bending, and heterogeneous phase evolution.

The temperature dependence of the domain-wall velocity originates from the explicit temperature dependence of both the Landau coefficient (thermodynamic term) and the kinetic coefficient (dynamic term). The Landau coefficient governs the curvature and depth of the double-well potential, thus controlling the free-energy difference between the FE and AFE states, while the kinetic coefficient describes the rate of polarization relaxation. As temperature decreases, the potential wells deepen, increasing the free-energy imbalance (Δf) and accelerating domain-wall motion; however, at lower temperatures, kinetic limitations dominate due to the saturation of L.

Using this multiscale model, we performed simulations of the FE-AFE phase transition under low-temperature conditions. Below the critical transition temperature, a local structural perturbation - such as a microcrack, inclusion, or dislocation - was introduced to initiate nucleation. As illustrated in Fig. 5a, the antiferroelectric phase preferentially nucleates at the perturbation site and subsequently propagates outward, forming a curved domain wall that advances with a diffusion-like velocity V. This process progressively transforms the ferroelectric phase into the antiferroelectric phase over time (Video S2).

Figure 5e presents the computed domain-wall velocity as a function of temperature, directly obtained from the nondimensionalized phase-field simulations and then converted to physical units through the kinetic coefficient described above. As shown in Fig. S7, the simulations reveal that the lattice-transformation energy barrier gradually decreases as temperature decreases, leading to a larger ΔG and a correspondingly stronger thermodynamic driving force. This explains the initial rapid increase of the phase-transition velocity at low temperatures (Fig. 5e). However, once the temperature falls below a critical threshold, the energy barrier becomes nearly eliminated, and the system approaches its minimum-energy configuration. In this regime, the domain-wall motion is no longer limited by the thermodynamic driving force but by intrinsic kinetic factors - such as structural rearrangement and polarization relaxation45,46 - which cause the kinetic coefficient to saturate. Consequently, the overall domain-wall velocity V reaches a limiting value Vlimit, representing the intrinsic dynamical limit of the lattice-transformation process rather than thermodynamic control.

Overall, these simulations demonstrate that the phase-field model quantitatively links first-principles energetics with mesoscale domain dynamics, clarifying how the mathematical framework governs the physical features observed in Fig. 5, including nucleation, defect-induced domain-wall curvature, and temperature-dependent interface velocity. The excellent agreement between the simulated and experimental data demonstrates that the self-consistent coupling of polarization, elastic, and thermal fields in the model successfully captures the intrinsic temperature-dependent kinetics of the FE-AFE interface migration.

Crucially, our work identifies and quantifies, for the first time, the theoretical kinetic limit of the FE–AFE phase transition driven by temperature. This limit corresponds to the minimum achievable transformation timescale dictated by intrinsic thermodynamic and elastic constraints of the material system. It provides a physically meaningful benchmark for the fastest possible response of ferroelectric materials driven by temperature and establishes a rigorous boundary for the design of ferroelectric devices.

Methods
Sample preparation
Pb(Zr0.97Ti0.03)O3-1%Nb2O5 (PZT97/3) ceramics were fabricated using a conventional solid-state reaction method. Stoichiometric amounts of analytical-grade PbO, ZrO2, TiO2, and Nb2O5 powders were weighed with an additional 1 wt.% PbO to compensate for volatilization loss. The powders were thoroughly mixed by ball milling in ethanol with zirconia media for 12 h, followed by drying and sieving. The resulting mixture was calcined at 850 °C for 3 h in air with a heating rate of 5 °C/min and a cooling rate of 10 °C/min to promote phase formation. The calcined powders were ground, pressed into pellets under a uniaxial pressure of 30 MPa, and sintered in covered alumina crucibles at 1200 °C for 2 h under identical heating and cooling rates to obtain dense ceramics. The sintered samples were polished and subsequently annealed at 400 °C for 1 h in air to relieve residual stress and ensure structural stability before electrical and structural characterization. In this study, 1 wt.% Nb2O5 was incorporated into Pb(Zr0.97Ti0.03)O3 to improve electrical performance. Previous reports indicate that Nb doping enhances electromechanical response while causing only minor shifts in phase boundaries without altering lattice symmetry47. Thus, Nb2O5 primarily stabilizes the perovskite structure while preserving the intrinsic phase-transition pathway of Zr-rich PZT.

Neutron diffraction
To investigate the evolution of the crystal structure of PZT97/3 ceramics with temperature, variable-temperature neutron diffraction experiments were conducted on the High-Resolution Neutron Diffractometer (HRND) at the China Academy of Engineering Physics. The sample was synthesized using the conventional solid-state reaction method. The neutron wavelength was set to 1.8838 Å. The sample was mounted in a high-precision cryogenic temperature control environment. Measurements were sequentially performed at the following temperature points: room temperature (25 oC), low temperature (−50 oC), and upon reheating back to 25 oC, to observe structural changes during both cooling and heating paths. At each temperature point, neutron diffraction patterns were collected over a 2θ range of 20o to 150o, with a minimum integration time of 6 hours to ensure sufficient statistical precision in the diffraction peak intensities. For the precise determination of crystal structure parameters, all diffraction data were refined using the Rietveld method via the FullProf program suite. During the refinement process, the background, polynomial peak shape function, zero-point error, lattice parameters, and atomic positional parameters were sequentially fitted. The initial structural models used were the ferroelectric phase (R3c) and the antiferroelectric phase (Pbam).

Transmission electron microscopy
Specimens for TEM were first mechanically thinned and then Ar+ ion–milled, with the ion-milling voltage stepped down from 3 keV to 0.5 keV to minimize beam damage. A thin carbon coat was applied to suppress charging under the electron beam. Bright-field (BF) images, dark-field (DF) images and selected-area electron diffraction (SAED) patterns were collected on a JEOL JEM‑F200 microscope. In-situ low-temperature TEM experiments were performed on a JEOL JEM‑F200 equipped with a Gatan model 636 cooling holder. High-angle annular dark-field (HAADF) imaging at atomic resolution was conducted on a Cs-corrected ThermoFisher Spectra 300 microscope with the accelerating voltage of 300 kV.

First-principles calculation
Our first-principles calculations were performed with the CASTEP code based on the density functional theory (DFT) using ultrasoft pseudopotentials and a plane-wave expansion of the wave functions. The lattice parameters of R3c, Pbam, PI are based on the refined structure of Rietveld and the TEM data. The exchange correlation potentials were described with the generalized gradient approximation (GGA). The kinetic energy cutoff of plane waves was set as 400 eV. For relaxation of structures, the Brillouin zone was sampled by Monkhorst-Pack grid grip with k-spacing of 0.2 Å−1. The values of the kinetic energy cutoff and the k-point grid were determined by ensuring the convergence of total energies within an accuracy of 1 meV/atom. The self-consistent field (SCF) tolerance of 5.0 × 10-5 eV/atom was used during structural relaxation via Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization. In order to investigate the free energy of AFE and FE structures under different temperatures, we apply the quasi-harmonic phonons model in which the nonequilibrium Gibbs function G can be written as follows:

G = U + PV − TS

Where T is temperature and F = U − TS is the Helmholtz free energy.

Phase field simulation
To describe the phase transition from ferroelectric (FE) to antiferroelectric (AFE) states which occurs in ferroelectric materials when the temperature falls below a critical threshold, the ferroelectric polarization (PFi) and antiferroelectric polarization (PAi) are commonly chosen as order parameters to characterize the degree of lattice polarization48. The total free energy density of the system comprises several contributions, including the standard Landau–Devonshire energy density, the elastic energy density, the polarization gradient energy density, and the electrostatic energy density, as well as the coupling energy density arising from the electrostrictive interactions between stress and the FE/AFE polarizations. Accordingly, the total free energy density is formulated as:

F = ∫ (flandau + felas + fgrad + felec + fcoup) dV (1.1)

The Landau energy density is formulated as49:

flandau = ∑ (αiPi2 + α11Pi4 + α111Pi6) + ∑ α12Pi2Pj2 + ∑ α112Pi4Pj2 + ∑ α123Pi2Pj2Pj2
+ ∑ (βiAi2 + β11Ai4 + β111Ai6) + ∑ β12Ai2Aj2 + ∑ β112Ai4Aj2 + ∑ β123Ai2Aj2Ak2
+ μ11 ∑ Pi2Ai2 + μ12 ∑ Pi2Aj2 + μ44 ∑ PiPjAiAj (1.2)

where the subscripts i, j denote Cartesian components (i, j = 1, 2, 3); μij accounts for the coupling between the FE and AFE polarizations; αi, αij, αijk, and βi, βij, βijk are the FE and AFE dielectric stiffnesses, respectively, under constant stress. The coefficients α1, β1 are related to temperature, which are determined by the Curie–Weiss law:

α1(T) = (T − T0)/2ε0C, β1(T) = (T − T0AFE)/2ε0CAFE (1.3)

where T is the temperature, ε0 is the vacuum permittivity, and C is the Curie constant. T0 and T0AFE represent the Curie-Weiss temperatures of the ferroelectric and antiferroelectric phases, respectively.

The elastic energy density is defined as:

f elas = 1/2 Cijkl eij ekl = 1/2 Cijkl (εij − ε0ij) (εkl − ε0kl) (1.4)

where Cijkl is the elastic stiffness tensor, and eij, εij and ε0ij denote the elastic strain, total strain and spontaneous strain, respectively. The spontaneous strain is defined by the electrostrictive coefficients Qijk and the FE and AFE polarizations. The total strain εij can be derived from the displacement field ui, εij = 1/2(ui,j + uj,i). In Fourier space, the elastic displacement field is solved by refs. 50,51:

u~i = iAik−1 ξl π~kl = ∑j Dij(ξ) (1.5)

where πeij = −Cijkl ε0kl, Xi = −iCijkl ε0kl ξj, i = −1. The wavy line on the symbol indicates its Fourier transform, and ξ is the wave vector. For cubic crystals, the explicit expressions for relevant quantities such as Nij(ξ) and D(ξ) take the forms51:

D(ξ) = μ2μ(λ + 2μ + μ0)ξ6 + μ02μ(3λ + 3μ + μ0)ξ2ξ2ξ2 + μ0(2λ + 2μ + μ0)ξ2(ξ2ξ2 + ξ2ξ2 + ξ2ξ2) (1.6)

Nij(ξ) = μ2ξ4 + θξ2(ξ2 + ξ2) + γξ2ξ2 (1.7)

Nij(ξ) = −(λ + μ)ξi ξj (μξ2 + μ0ξ2) (1.8)

where ξ2 = ξi ξi, θ = μ(λ + μ + μ0), γ = μ0(2λ + 2μ + μ0), λ = c12, μ = c44, μ0 = c11 − c12 − 2c44. cij is the elastic modulus of the material. According to Ginzburg–Landau theory, the free energy also includes the spatial gradient of the order parameters, which accounts for the domain wall energy in systems. For simplicity, the lowest-order gradient energy density is given by:

fgrad = 1/2 GF11 (PFi,12 + PFi,22 + PFi,32) + GF12 (PFi,1 PFi,2 + PFi,2 PFi,3 + PFi,3 PFi,1) + 1/2 GF44 (PFi,2 + PFi,1)2 + (PFi,3 + PFi,2)2 + (PFi,1 + PFi,3)2 + 1/2 GF044 (PFi,2 − PFi,1)2 + (PFi,3 − PFi,2)2 + (PFi,1 − PFi,3)2 + 1/2 GA11 (PA1,12 + PA2,22 + PA3,32) + GA12 (PA1,1 PA2,2 + PA2,2 PA3,3 + PA3,3 PA1,1) + 1/2 GA44 (PA1,2 + PA2,1)2 + (PA2,3 + PA3,2)2 + (PA1,3 + PA3,1)2 + 1/2 GA044 (PA1,2 − PA2,1)2 + (PA2,3 − PA3,2)2 + (PA1,3 − PA3,1)2 (1.9)

where GF11, GF12, GF44, GF044 and GA11, GA12, GA44, GA044 are the gradient energy coefficients for FE and AFE polarizations, respectively. PFi,j / PAi,j denotes the derivative of the ith component of the polarization vector, PFi=Ai, with respect to the jth coordinate.

The electrical energy density is expressed as52:

felec = − 1/2 κij Ei Ej − Pi Ei (1.10)

where the total electric field, Ei, has two contributions, the depolarization field, Edi, and the external electric field, Eexti, such that: Ei = Edi + Eexti, κij is the dielectric tensor. The depolarization field Edi in Fourier space can be written as53:

E~di = −iξi ϕ~ = ξi (ξ1 P~1 + ξ2 P~2 + ξ3 P~3) / (κ11 ξ12 + κ22 ξ22 + κ33 ξ32) (1.11)

The expression for the coupling energy density between ferroelectric and antiferroelectric polarization and stress σij is given by:

fcoup = −Q11 PFi2 σii − Q12 PFi2 σjj − 1/2 Q44 PFi Pj σij − Z11 PAi2 σii − Q12 PAi2 σjj − 1/2 Z44 PAi PAj σij (1.12)

As noted above, the elastic strain energy depends on both the polarization state and applied mechanical load. An applied electric field can modify the polarization, altering the strain energy; conversely, external stress can affect the polarization, and hence influence the electrostatic energy. The temporal evolution of the polarization fields under applied electric and/or mechanical loads is governed by the time-dependent Ginzburg–Landau (TDGL) equation54,55:

∂PFi / ∂t = −L (δF / δPFi) (1.13)

∂PAi / ∂t = −L (δF / δPAi) (1.14)

where LFE=AFE is the kinetic coefficient, F is the total free energy, and δF/δP is the thermodynamic driving force for the polarization evolution, t denotes time. The semi-implicit Fourier-spectral method was employed to solve the partial differential Eqs. (1.13) and (1.14) in the present work.

For computational efficiency and generality, the following set of dimensionless variables is adopted:

r* = r/r0, t* = |α0|Lt, P* = P/P0, α1* = α1/|α0|, α11* = α11/|α0|P02, α12* = α12/|α0|P02, α111* = α111/|α0|P04, α112* = α112/|α0|P04, α123* = α123/|α0|P04, β1* = β1/|α0|, β11* = β11/|α0|P02, β12* = β12/|α0|P02, β111* = β111/|α0|P04, β112* = β112/|α0|P04, β123* = β123/|α0|P04, μ11* = μ11/|α0|P02, μ12* = μ12/|α0|P02, μ44* = μ44/|α0|P02, Q11* = Q11 P02, Q12* = Q12 P02, Q44* = Q44 P02, Z11* = Z11 P02, Z12* = Z12 P02, Z44* = Z44 P02, E* = (√|α0|/P0) E, σ* = (1/|α0|P02) σ, κ* = |α0| κ, c11* = c11/|α0|P02, c12* = c12/|α0|P02, c44* = c44/|α0|P02, GF11* = GF11/G0, GF12* = GF12/G0, GF44* = GF44/G0, GF044* = GF044/G0, GA11* = GA11/G0, GA12* = GA12/G0, GA44* = GA44/G0, GA044* = GA044/G0 (1.15)

where P0 = 0.4 Cm−2 is the spontaneous polarization at room temperature, α0 = (22−479) 3.8 × 105 m2N/C2, and G0 = 1.73 × 10−10 m4N/C2 is a reference value for the gradient energy coefficient.

In this work, the simulations began with an initially mono-domain ferroelectric state, characterized by assigning PFi = 1 and PAi = 0. A total of 256 × 256 discrete grid points were employed with a cell size of Δx* = Δy* = 1, and periodic boundary conditions were imposed in both the x and y directions. These conditions emulate an infinite single crystal, eliminating edge effects and isolating intrinsic material behavior.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "“Diffusional dynamic process of irreversible ferroelectric to antiferroelectric phase transition and the rate limit” by Minyue Ge,\nBing Han, Zhengqian Fu et al.\nI found the subject of this manuscript particularly interesting, as it addresses the evidence of a ferroelectric–antiferroelectric\n(FE–AFE) phase transition using powerful experimental techniques such as neutron diffraction and TEM, complemented by\ntheoretical considerations. In my opinion, the way the authors capture the diffusional dynamics during the phase transition in\nreal time is especially valuable. Equally intriguing is the finding that, once the transition to the AFE phase occurs at low\ntemperature, this phase remains irreversible and is still present at higher temperatures. These aspects make the study both\ntimely and relevant. In the following, I will provide several comments and questions that arise while reading the manuscript.\n1. I assume the study was performed on ceramics rather than thin films. In any case, details about this aspect, or about how\nthe samples were prepared, are missing.\n2. In the introduction, the authors mention Pb(Zr₀.₉₇Ti₀.₀₃)O₃–1%Nb₂O₅ (PZT97/3). What is the role of Nb₂O₅? Could this\nadditive influence phase stabilization, diffusion velocity, etc.?\n3. Referring to “The in-situ videos, for the first time, revealed that the FE–AFE transition exhibits relaxation behavior and is a\ndiffusive dynamic process.” Could this be correlated with other real-time recorded transitions in ferroelectrics, for example\nthose observed under electric fields? Aren’t those also diffusive and exhibiting relaxation? Could such a description be\ncharacteristic of phase transitions in general?\n4. What is the reason for the unusually long duration of the phase transition reported in this manuscript? Which physical\nmechanism drives or limits this transition?\n5. Regarding “The conventional phase diagram for PZT materials depicts a transition from the paraelectric phase (cubic) to\nthe ferroelectric (R3m/R3c) phase upon cooling (Fig. 1e). However, our experiments further reveal an additional FE–AFE\nphase transition occurring in the low-temperature region (phase boundary shown as a red dashed line in Fig. 1e). This\nfinding expands and updates the established PZT phase diagram.” Is this truly a new concept, given that many PZT phase\ndiagrams already suggest a transition between FE (R3c) and AFE (Pbam) for compositions with a large Zr/Ti ratio at low\ntemperatures? See, for example, Figure 5 in https://iopscience.iop.org/article/10.1088/0022-3727/49/27/275305.\n6. There also seem to be some inaccuracies in the text. For instance: “The ferroelectric to antiferroelectric (FE–AFE) phase\ntransition involves crystal symmetry breaking.” The FE→AFE transition does not involve symmetry breaking; rather, it\ntypically corresponds to symmetry restoration or an increase in symmetry (from a lower-symmetry polar phase to a higher-\nsymmetry antipolar phase).", "reviewer_2": "Review of “Diffusional dynamic process of irreversible ferroelectric to antiferroelectric phase transition and the rate limit”\nThe manuscript starts with an introduction to Ferroelectric (FE) to Antiferroelectric (AFE) phase transitions, and motivates the\nneed to study them. Historically, such transitions have been studied statically with TEM, neutron diffraction, and other\nmethods, and dynamically with mainly electrical methods. In this work, such static and dynamic measurements are done to\nstudy the phase transition, but a new method is employed to dynamically study the phase transition: in-situ TEM videos. The\nexperimental results are put into context and explained using DFT simulations, and a phase-field model is developed to\ndescribe the dynamics as well.\nDue to the extensive experimental results and novel experimental methodology, as well as the comprehensive\ntheoretical/simulation work done to put the experimental results in context, I recommend this paper for eventual publication\nin Nature Communications. The experimental work is convincing, and the theoretical work will act as a good guideline to\nunderstanding the FE-AFE transitions of new materials. However, there remain several key questions to be answered, and\nsome changes to the manuscript are in order before the manuscript is accepted. The questions are as follows:\n1. In Neutron diffraction experiments, the peaks that show up upon cooling, are\nassigned to be ¼(161), ¼(-160), and ¼(152). From these peaks, it is concluded that the structure changes from the R3c phase to the Pbam phase.\na. Are these the only new peaks we would expect to see in the Pbam phase from the R3c phase? I would expect some other\npeaks to arise too. What does the structure factor predict for R3c and Pbam?\nb. The magnitudes of these peaks are quite small compared to the other R3c peaks. Why is that?\nc. What would the Rwp be for the middle and right plot in Fig. S1 if they were fit to the R3c phase instead of the Pbam\nphase? Similarly, what would the Rwp be for the left plot in Fig. S1 if it was fit to the Pbam phase? These comparisons are\nimportant for justifying the conclusions.\nd. Do the neutron diffraction patterns indicate a full switch from all R3c phase to all Pbam phase, or is there likely to be a\nmixture of the two phases in the sample at -50 and 25C respectively?\n2. If it takes 50 seconds for the domain wall diffusion to occur, why is a 2-hr relaxation necessary for the TEM measurement\n(line 149)? Is the sample being imaged under TEM for the whole 2 hrs? If not, how do you know when to start recording the\nTEM video?\na. My concern would be that the phase transition could be due to damage from the electron beam itself, or the neutron\nbeam in the neutron experiments for example. Are any experiments done to control for possible damage?\n3. The plots in Fig. 4 and Fig. S7 imply that at high temperatures, the R3c phase is preferred and the R3c to Pbam energy\nbarrier is low. Theoretically, what would be the temperature at which this phase transition is likely to happen within the\nwindow of the measurement?\na. It seems from the theory that this phase transition is not irreversible, but hysteretic. The data would be strengthened\ngreatly by observing the AFE-FE phase transition, and reporting the temperature of it. Is my understanding correct, and is\nthis a possible measurement?\nb. It would be interesting to compare a theoretically derived hysteresis for the phase of the material and compare it to an\nexperimentally verified hysteresis, and would greatly strengthen the results of the paper.\n4. Line 272-273 states that the intervention of external energy is necessary for the AFE-FE transition. Is this done to the\nsamples in question, and confirmed? It would strengthen the results to see this done. For example, could Fig. S3 be related\nto this claim?\n5. From line 284 – 285: how do the Gibbs free energy and the DFT calculations get incorporated into the phase field\nsimulations? It is not clear from the methods section or main text.\n6. In general, I find that the phase field simulation work is not well-explained and not put into context. The results are very\ninteresting: namely figure 5e, but I do not understand how the data in figure 5e comes from the phase field analysis in the\nmethods section. What parameters are fitted for? Which parameters are simulated? Which parameters, in the long\ndescription of the phase field method and equations, are the most important for estimating the domain wall velocity and the\ntemperature dependency?\na. More time should be spent describing figure 5e: how does the phase field prediction do such a good job of modeling the\nsystem? Unless I missed something, the only empirical parameters used in the simulations were the lattice sizes of the\ncrystals in question…\n7. Line 298 – 299: “This behavior [of the velocity saturating at 8.05 nm/s] is attributed to the reduction in the lattice\ntransformation barrier at lower temperatures.” How is this conclusion drawn, and what part of the phase field simulations or\nequations lead us to that conclusion?\n8. In general, the phase-field section of the methods section is very poorly connected to the main text. There is a lot of\nmath and work that goes into it, but there is little indication of how that math comes together to create Figure 5. For example,\na. What drives the transition?\nb. What parameters drive the diffusion of the domain wall?\nc. How is the microstructure defect implemented?\nd. What parameters are responsible for the temperature dependency of the velocity?\ne. Where does the DFT work fit into the phase-field simulations?\n9. Finally, the English language and grammar of the paper could use some work. I would be happy to suggest changes upon\nreceival of the manuscript modified with the questions above resolved."}, "author_response": "Response: We sincerely appreciate the reviewer’s thoughtful and encouraging\nevaluation of our work. We have carefully considered all of the reviewer’s constructive\ncomments and questions. In the following, we provide detailed, point-by-point\nresponses and have revised the manuscript accordingly to further improve its clarity,\ncompleteness, and overall quality.\n1. I assume the study was performed on ceramics rather than thin films. In any case,\ndetails about this aspect, or about how the samples were prepared, are missing.\nResponse: We highly appreciate the reviewer for carefully reading ours work. In this\nstudy, Pb(Zr Ti )O 1%Nb O (PZT97/3) ceramics were synthesized via\n0.97 0.03 3 2 5\nconventional solid-state sintering method, as shown in Fig. R1.\nAnalytical-grade PbO (≥99.9%), ZrO (≥99.9%), TiO (≥99.9%), and Nb O (≥99.9%)\n2 2 2 5\npowders were weighed according to the stoichiometric ratios. To compensate for lead\nvolatilization during high-temperature sintering, an additional 1-2 wt.% of PbO was\nintroduced. The weighed powders were mixed and homogenized using a planetary ball\nmill in ethanol with zirconia media at a ball-to-powder ratio of 10:1. The milling was\nperformed at 300 rpm for 12 h to ensure complete dispersion and uniform mixing of all\nconstituent oxides. The resulting slurry was dried at 80 ℃ and gently sieved through a\n100-mesh stainless-steel sieve to remove agglomerates and ensure a uniform particle\nsize distribution.\nThe mixed and sieved powders were subsequently calcined in an alumina crucible at\n850 ℃ for 2-4 h in air, with a controlled heating rate of 5 ℃/min and natural cooling\nto room temperature. This calcination step promoted solid-state diffusion and reaction\namong the oxides, forming a homogeneous perovskite PZT phase while minimizing\nPbO volatilization. The calcined powders were lightly crushed and subjected to a\nsecond round of ball milling in ethanol for 6 h to further reduce particle size and\nimprove compositional uniformity. After drying at 80 ℃, the powders were sieved\nthrough a 200-mesh screen to obtain fine precursor powders suitable for pressing.\nThe prepared powders were uniaxially pressed into disk-shaped pellets (typically 12\nmm in diameter and 1.0-1.5 mm thick) under a pressure of approximately 30 MPa using\na stainless-steel die. To prevent cracking and ensure homogeneous density, the green\ncompacts were preheated at 500 ℃ for 1 h to remove organic binders and absorbed\nmoisture. Subsequently, the pellets were sintered in covered high-purity alumina\ncrucibles at 1200 ℃ for 2-3 h in air, using a heating rate of 5 ℃/min and allowing\nnatural cooling to room temperature. The use of covered crucibles effectively\nsuppressed PbO evaporation and maintained stoichiometric balance in the ceramics.\nAfter sintering, the ceramics exhibited dense and uniform microstructures with relative\ndensities exceeding 95% of the theoretical value. The sintered pellets were polished on\nboth sides using progressively finer SiC abrasive papers, followed by ultrasonic\ncleaning in ethanol to remove any surface residues. Prior to electrical and structural\ncharacterization, the samples were annealed at 400 ℃ for 1 h in air to release residual\nstresses introduced during polishing and to stabilize the ferroelectric domain\nconfiguration.\nOverall, this solid-state sintering route enables the fabrication of high-quality PZT97/3\nceramics with well-developed perovskite grains, minimal secondary phases, and\nexcellent structural integrity. The carefully controlled heating and cooling profiles,\ncombined with optimized PbO compensation and covered crucible sintering, are\nessential for achieving reproducible microstructures and stable phase compositions\nsuitable for subsequent neutron diffraction and dielectric characterization.\nFig. R1| Solid-phase sintering flow chart.\nAction: We have added following discussions to the revised manuscript. (Please see\nthe parts marked by red color on page 15 of the revised manuscript)\nExperiments and Methods\nSample preparation\nPb(Zr Ti )O -1%Nb O (PZT97/3) ceramics were fabricated using a conventional\n0.97 0.03 3 2 5\nsolid-state reaction method. Stoichiometric amounts of analytical-grade PbO, ZrO ,\n2 2 5\nfor volatilization loss. The powders were thoroughly mixed by ball milling in ethanol\nwith zirconia media for 12 h, followed by drying and sieving. The resulting mixture\nwas calcined at 850 ℃ for 3 h in air with a heating rate of 5 ℃/min and a cooling rate\nof 10 ℃/min to promote phase formation. The calcined powders were ground, pressed\ninto pellets under a uniaxial pressure of 30 MPa, and sintered in covered alumina\ncrucibles at 1200 ℃ for 2 h under identical heating and cooling rates to obtain dense\nceramics. The sintered samples were polished and subsequently annealed at 400 ℃ for\n1 h in air to relieve residual stress and ensure structural stability before electrical and\nstructural characterization.\n2. In the introduction, the authors mention Pb(Zr Ti )O -1%Nb O (PZT97/3).\n0.97 0.03 3 2 5\nWhat is the role of Nb O ? Could this additive influence phase stabilization, diffusion\n2 5\nvelocity, etc.?\nResponse: We appreciate the reviewer’s insightful question. Previous studies have\nshown that excess Nb O positively promote sintering and improve electrical insulation\n2 5\n[1-3]. In addition, Noheda et al. [4] demonstrated through pulsed neutron diffraction\nthat Nb doping can also shift phase position in phase diagram, without altering the\nlattice symmetry. Fig. R2 compares the phase diagram of the PZT system containing 1\nwt.% Nb O with that of undoped PZT. The two diagrams are almost identical, with\n2 5\nonly slight shifts in the phase positions. Specifically, the boundaries between the low-\ntemperature ferroelectric (F ), high-temperature ferroelectric (F ), and paraelectric\nL H\ncubic (P ) phases remain nearly unchanged. This indicates that Nb O doping does not\nC 2 5\ndisrupt the intrinsic phase transition behavior of Zr-rich PZT.\n[Figure Redacted]\nFig. R2| The comparison of phase diagrams of undoped PZT and PZT+1%Nb O\n2 5\nfrom Ref. [4].\nReferences:\n[1] K.W. Lee, D.G. Lee, H.C. Song, S.M. Lim. Microstructure and Piezoelectric Properties of PMN-\nPAN-PZT Ceramics. J. Korean Inst. Electr. Electron. Mater. Eng 38 174-178 (2025).\n[2] X. H. Dai, Y. L. Wang. Multistage heat-electric energy conversion working on F - F phase\nR(LT) R(HT)\ntransition in nb doped PZT97/3 ceramics. Ferroelectric. 109 253-258 (1990).\n[3] D.Z. Sun, C. F. Qu, C.H. Yao, H. Du, Q. H. Jin, S. W. Lin. Relation between the phase transitional\nbehavior and the content of additive niobium oxide in PZT 97/3. Ferroelectric. 262 281-286 (2001).\n[4] B. Noheda, J. A. Gonzalo, M. Hagen. Pulsed neutron diffraction study of Zr-rich PZT. J. Phys.:\nCondens. Matter 11 3959-3965 (1999).\nAction: We have added following discussions to the revised manuscript. (Please see\nthe parts marked by red color on page 15 of the revised manuscript)\nThe sintered samples were polished and subsequently annealed at 400 ℃ for 1 h in air\nto relieve residual stress and ensure structural stability before electrical and structural\ncharacterization. In this study, 1 wt.% Nb O was incorporated into Pb(Zr Ti )O\n2 5 0.97 0.03 3\nto improve electrical performance. Previous reports indicate that Nb doping enhances\nelectromechanical response while causing only minor shifts in phase boundaries\nwithout altering lattice symmetry [47]. Thus, Nb O primarily stabilizes the perovskite\n2 5\nstructure while preserving the intrinsic phase-transition pathway of Zr-rich PZT.\n3. Referring to “The in-situ videos, for the first time, revealed that the FE-AFE\ntransition exhibits relaxation behavior and is a diffusive dynamic process.” Could this\nbe correlated with other real-time recorded transitions in ferroelectrics, for example\nthose observed under electric fields? Aren’t those also diffusive and exhibiting\nrelaxation? Could such a description be characteristic of phase transitions in general?\nResponse: We sincerely thank the reviewer for this insightful comment. While such\nobservations of ferroelectric phase transitions remain relatively rare, similar real-time\nprocesses mainly focus on electric field induced domain switching. It is interesting to\nfind that the behavior of domain switching is highly sensitive to experimental\nconditions such as contact mode. As shown in Fig. R3, the domain switching is a sharp\nprocess if the PZT film has both top and bottom electrode while it is a diffuse process\nif the PZT film only has bottom electrode.\nOur in-situ observations reveal that the FE-AFE transition proceeds through a diffusive\nprocess, which is mainly because the FE and AFE phase have small difference in free\nenergy (<0.1 eV, Fig. 4a). Thus, we speculate that similar relaxation and diffusive\ndynamics could also occur in other types of ferroelectric transitions, particularly when\ncompeting phases are energetically close. Further systematic studies will be required to\nvalidate this hypothesis and to extend our understanding of similar dynamics in other\nferroelectric transitions.\n[Figure Redacted]\nFig. R3| Real-time TEM observations of the bias-induced domain switching\nbehaviors in PZT thin film with different electrical contact configuration. (a)\nDomain switching behaviors in PZT with both top and bottom electrodes, where almost\nthe entire domain switches abruptly at -5 V. (b) Domain switching behaviors in PZT\nwith only bottom electrode. The video snapshot shown here is adapted from the\nSupporting Information of Ref. [5].\nReference:\n[5] Y. Zhang, L. Xie, J. Kim, A. Stern, H. Wang, K. Zhang, X. Yan, L. Li, H. Liu, G. Zhao, H. Chi, C.\nGadre, Q. Lin, Y. Zhou, C. Uher, T. Chen, Y. H. Chu, J. Xia, R. Wu, X. Pan. Discovery of a magnetic\nconductive interface in PbZr Ti O /SrTiO heterostructures. Nat. Commun. 9 685 (2018).\n0.2 0.8 3 3\n4. What is the reason for the unusually long duration of the phase transition reported\nin this manuscript? Which physical mechanism drives or limits this transition?\nResponse: We thank the reviewer for this important question. Although the FE-AFE\ntransition in PZT is conventionally considered rapid, our in-situ TEM observations (Fig.\n2g-l) show that, in PZT97/3 at -50 °C, the transformation lasts ~50 s. Through TEM\ndata, we discovered and confirmed the phenomenon that the FE-AFE phase transition\ntime of PZT97/3 is extremely long. The extremely long transformation time of the FE-\nAFE phase transition is due to the fact that the phase boundary migration is jointly\nrestricted by the interface strain type and the obstruction of the domain wall, resulting\nin a slow diffusion-driven mechanism. Firstly, as shown in Fig. 3d, lattice fluctuation\nin the BP region (~40 pm) is much smaller than that of the AFE phase (~80 pm),\nindicating that the transformation requires overcoming a substantial interfacial strain\nenergy. The large lattice fluctuation indicates a significant structural difference, so the\nphase transition requires a larger displacement and thus takes a longer time for\nrearrangement. Secondly, Time-resolved TEM (Figs. 2g-2l) demonstrates that the AFE\ntransformation front is intermittently pinned by pre-existing FE domain walls. The\nobstruction of this FE domain wall will further slowdown the rate of the FE-AFE phase\ntransformation. Finally, phase-field-fitted (Fig. 5) show that the AFE front velocity is\nlimited to ~8.05 nm/s and exhibits diffusion-type behavior, in good agreement with the\nexperimental observations. These phase-field-fitted results further demonstrate that the\nFE-AFE phase transition is a long-duration process. In summary, the unusually long\nFE-AFE transition time originates from a larger displacement and the obstruction of the\ndomain wall.\n5. Regarding “The conventional phase diagram for PZT materials depicts a transition\nfrom the paraelectric phase (cubic) to the ferroelectric (R3m/R3c) phase upon cooling\n(Fig. 1e). However, our experiments further reveal an additional FE-AFE phase\ntransition occurring in the low-temperature region (phase boundary shown as a red\ndashed line in Fig. 1e). This finding expands and updates the established PZT phase\ndiagram.” Is this truly a new concept, given that many PZT phase diagrams already\nsuggest a transition between FE (R3c) and AFE (Pbam) for compositions with a large\nZr/Ti ratio at low temperatures? See, for example, Figure 5 in\nhttps://iopscience.iop.org/article/10.1088/0022-3727/49/27/275305.\nResponse: We appreciate the reviewer’s valuable comment. Although Li et al. [6] have\nsuggested a possible FE-AFE phase transition occurring in the low-temperature region\nby theoretical calculations and optical analyses, they didn’t give any structural evidence\n(Fig. R4). In our work, we confirmed the FE-AFE transition at low temperature using\nlow-temperature neutron diffraction, as shown in Fig. R5. We observed that upon\ncooling from 25 °C to -50 °C, new diffraction peaks characteristic of the AFE (Pbam)\nphase emerged in the neutron diffraction patterns of PZT ceramics, demonstrating that\nthe material adopts an FE-AFE transition at low temperature. Therefore, while our\nstudy does not propose an entirely new PZT phase diagram, it experimentally validates,\nrefines, and extends the previously hypothesized phase relationships by providing\nstructural evidence for the FE-AFE transition at low temperatures.\n[Figure Redacted]\nFig. R4| The phase diagram of PZT ceramics derived from the thermal evolution\nof the transition energies with the temperature and Ti compositions [6].\nFig. R5| FE-AFE phase transition of PZT97/3 ceramics driven by temperature.\nThe neutron diffraction patterns of the PZT97/3 ceramics at 25 ℃, -50 ℃, and 25 ℃,\nrespectively.\nReference:\n[6] M. J. Li, L. P. Xu, K. Shi, J. Z. Zhang, X. F. Chen, Z. G. Hu, X. L. Dong, J. H. Chu. Interband\nelectronic transitions and phase diagram of PbZr TiO (0.05 ≤ x ≤ 0.70) ceramics: ellipsometric\n1−x x 3\nexperiment and first-principles theory. J. Phys. D: Appl. Phys. 49 275305 (2016).\nWe have added following discussions to the revised manuscript. (Please see the parts\nmarked by red color on page 5 of the revised manuscript)\nThis trend is closely related to lattice reconstruction and enhanced structural ordering.\nFor piezoelectric ceramic materials, the conventional phase diagram indicates that\nduring the cooling process from 300 K to 200 K, they will transform from the\nantiferroelectric phase to the ferroelectric phase (Fig. 1e). However, theoretical\ncalculations and optical analyses by Li et al. [33] suggest an additional FE-AFE phase\ntransition occurring in the low-temperature region. In this work, we demonstrate that\nsuch a phase transition indeed exists by undoubted experimental evidence. This\ndiscovery has been experimentally verified to confirm that the update of the PZT phase\ndiagram by Li et al. is correct. As shown in Fig. 1(e), we have redrawn the updated\nphase diagram. Fig. 1f-h depicts the unit cells change during the FE-AFE transformation,\nwhich is manifested as the displacement of Pb atoms along the a-axis direction, giving\nrise to the antiferroelectric phase.\n6. There also seem to be some inaccuracies in the text. For instance: “The ferroelectric\nto antiferroelectric (FE-AFE) phase transition involves crystal symmetry breaking.”\nThe FE-AFE transition does not involve symmetry breaking; rather, it typically\ncorresponds to symmetry restoration or an increase in symmetry (from a lower-\nsymmetry polar phase to a higher-symmetry antipolar phase).\nResponse: We sincerely thank the reviewer for the careful reading and valuable\ncomment. We acknowledge that there was an error in the original description. Indeed,\nthe antiferroelectric-to-ferroelectric (AFE-FE) phase transition is the process involving\ncrystal symmetry breaking, rather than the reverse.\nWe have added following discussions to the revised manuscript. (Please see the parts\nmarked by red color on page 3 of the revised manuscript)\nThe ferroelectric to antiferroelectric (FE-AFE) phase transition involves crystal\nsymmetry changing, dipole reordering, and significant changes in energy and entropy,\nrepresenting a frontier interdisciplinary field in condensed matter physics and materials\nscience [1-5].\nReviewer #2 (Remarks to the Author):\nThe manuscript starts with an introduction to Ferroelectric (FE) to Antiferroelectric\n(AFE) phase transitions, and motivates the need to study them. Historically, such\ntransitions have been studied statically with TEM, neutron diffraction, and other\nmethods, and dynamically with mainly electrical methods. In this work, such static and\ndynamic measurements are done to study the phase transition, but a new method is\nemployed to dynamically study the phase transition: in-situ TEM videos. The\nexperimental results are put into context and explained using DFT simulations, and a\nphase-field model is developed to describe the dynamics as well.\nDue to the extensive experimental results and novel experimental methodology, as well\nas the comprehensive theoretical/simulation work done to put the experimental results\nin context, I recommend this paper for eventual publication in Nature Communications.\nThe experimental work is convincing, and the theoretical work will act as a good\nguideline to understanding the FE-AFE transitions of new materials. However, there\nremain several key questions to be answered, and some changes to the manuscript are\nin order before the manuscript is accepted.\nResponse: We sincerely thank the reviewer for the positive and insightful evaluation\nof our manuscript. We have carefully reviewed all the reviewer’s constructive\ncomments and questions. In the following, we provide detailed point-by-point\nresponses and have revised the manuscript accordingly to further enhance its scientific\nclarity and completeness.\n1. In Neutron diffraction experiments, the peaks that show up upon cooling, are\nassigned to be ¼(161), ¼(-160), and ¼(152). From these peaks, it is concluded that the\nstructure changes from the R3c phase to the Pbam phase.\na. Are these the only new peaks we would expect to see in the Pbam phase from the R3c\nphase? I would expect some other peaks to arise too. What does the structure factor\npredict for R3c and Pbam?\nResponse: We thank the reviewer for this insightful question. We used FullProf to\nsimulate the neutron diffraction patterns of R3c and Pbam, and it was found the three\nreflections ¼(161), ¼(-160), and ¼(152) are not the only new peaks that may appear\nduring the transition from the R3c phase to the Pbam phase(Fig. R6). As shown in Fig.\nR7, the Pbam phase is featured with antipolar displacements of Pb2+ cations and the\nantiphase tilting of oxygen octahedra(a-a-c0). This emergence of characteristic\nsuperlattice reflections such as ¼(161), ¼(-160), and ¼(152) in Pbam are originated\nfrom antipolar displacements of Pb2+ cations. Because neutron diffraction is sensitive\nto oxygen atoms, the 1/2{ooo}-type (The “o” refers to odd indices) superlattice\nreflection originated from antiphase oxygen octahedral tilting can also be observed.\nWe just use the relatively strong reflections of ¼(161), ¼(-160), and ¼(152) to indicate\nthe occurrence of the R3c-Pbam phase transition.\nFig. R6| Simulated neutron diffraction patterns of the R3c and Pbam phases.\nFig. R7| The Schematic illustration of the crystal structures of the Pbam phases.\nb. The magnitudes of these peaks are quite small compared to the other R3c peaks. Why\nis that?\nResponse: As shown in Fig. R9, the R3c phase is characterized by parallel\ndisplacements of Pb2+ cations and antiphase tilting of oxygen octahedra (a-a-a-),\nwhile the Pbam phase is featured with antipolar displacements of Pb2+ cations and the\nantiphase tilting of oxygen octahedra(a-a-c0). This emergence of superlattice reflections\nsuch as ¼(161), ¼(-160), and ¼(152) in Pbam are originated from antipolar\ndisplacements of Pb²⁺ cations while the superlattice reflections in R3c come from the\nantiphase tilting of oxygen octahedra. Because neutron diffraction is sensitive to\noxygen atoms, the fourfold superlattice peaks in Pbam phase are quite small compared\nto the superlattice peaks induced by antiphase tilting of oxygen octahedra in R3c phase.\nFig. R9| The Schematic illustration of the crystal structures of the (a) R3c and (b)\nPbam phases.\nc. What would the Rwp be for the middle and right plot in Fig. S1 if they were fit to the\nR3c phase instead of the Pbam phase? Similarly, what would the Rwp be for the left\nplot in Fig. S1 if it was fit to the Pbam phase? These comparisons are important for\njustifying the conclusions.\nResponse: We sincerely thank the reviewer for this valuable and insightful question.\nTo quantitatively assess the phase identification, we re-performed Rietveld refinements\nof the neutron diffraction data at 25 °C, -50 °C, and re-heated 25 °C using both R3c and\nPbam structural models. The refined results are summarized in Fig. R10. For the initial\nsample at 25 °C, the R3c model yields Rwp = 7.19%, whereas the Pbam model gives a\nmuch higher Rwp = 32.6%, indicating that PZT97/3 has a R3c structure at room\ntemperature. Upon cooling to -50 °C, the R3c model results in Rwp = 20.3%, while the\nPbam model fits significantly better with Rwp = 7.89%, confirming the R3c-Pbam\ncompletely phase transition at -50 °C. When the PZT97/3 is heated to 25 °C, the Pbam\nmodel gives Rwp = 7.69%, compared with Rwp = 19.0% for the R3c model, suggesting\nthat the Pbam phase remains stable upon returning to room temperature without an\nexternal electric field. These comparisons of Rwp values across different temperatures\nprovide direct evidence for the temperature-induced irreversible FE (R3c)-AFE (Pbam)\nphase transition in PZT97/3 ceramics.\nFig. R10| Rietveld refinement of synchrotron neutron diffraction data for PZT97/3\nunder selected temperatures.\nWe have added figure to the revised Supplemental material. (Please see the parts\nmarked by red color on page 2 of the revised Supplemental material)\nFig. S1. Rietveld refinement of synchrotron neutron diffraction data for PZT97/3 under\nselected temperatures.\nd. Do the neutron diffraction patterns indicate a full switch from all R3c phase to all\nPbam phase, or is there likely to be a mixture of the two phases in the sample at -50 ℃\nand 25 ℃ respectively?\nResponse: We thank the reviewer for this insightful question. To clarify the possible\ncoexistence of the R3c and Pbam phases, we performed additional Rietveld refinements\nof the neutron diffraction data at -50 °C and 25 °C by including both R3c and Pbam\nstructural models. The results are summarized in Fig. R11. At -50 °C, the refined phase\nfractions are 98.17% Pbam and 1.83% R3c, while at 25 °C they are 95.42% Pbam and\n4.58% R3c. At -50 ℃ and 25 ℃, the refined phase fractions of the second phase were\nboth less than 5%. As show in Figs. R10 and R11, compared with the single-phase\nPbam refinement (Rwp = 7.69%) at 25 ℃, the two-phase refinements show slightly\nhigher residuals (Rwp = 8.47%) at 25 ℃. Then compared with the single-phase Pbam\nrefinement (Rwp = 7.89%) at -50 ℃, the two-phase refinements show slightly higher\nresiduals (Rwp = 8.93%) at -50 ℃. Since the proportion of the second phase is less than\n5% and the introduction of the second phase has increased the Rwp, we can conclude\nthat at -50 ℃ and 25 ℃, the two phases do not coexist but are in a single-phase state.\nFig. R11| Rietveld refinement of synchrotron neutron diffraction data for PZT97/3\nunder selected temperatures.\n2. If it takes 50 seconds for the domain wall diffusion to occur, why is a 2-hour\nrelaxation necessary for the TEM measurement (line 149)? Is the sample being imaged\nunder TEM for the whole 2 hours? If not, how do you know when to start recording the\nTEM video?\na. My concern would be that the phase transition could be due to damage from the\nelectron beam itself, or the neutron beam in the neutron experiments for example. Are\nany experiments done to control for possible damage?\nResponse: We sincerely thank the reviewer for the valuable and professional comments.\nAfter cooling the sample to -50 °C, nucleation of the AFE phase requires approximately\ntwo hours, as the system needs to overcome the associated energy barrier. During this\nwaiting period, the sample was continuously recorded under TEM. For clarity and\nconciseness, representative frames at the initial state and at approximately 30 minutes,\n1 hour, and 1.5 hours are shown (Fig. R12). No noticeable changes in the domain\nconfiguration were observed throughout this period, suggesting that the AFE phase had\nnot yet nucleated. Once nucleation occurs, the FE-AFE transition and the associated\ndomain wall motion occur on a timescale of seconds, completing within about 50\nseconds.\nTo address potential concerns regarding beam-induced effects, we performed control\nexperiments at room temperature under identical imaging conditions. Even after\ncontinuous electron-beam irradiation for approximately two hours, the domain structure\nand the phase structure remained unchanged (Fig. R13), confirming the structural\nstability of PZT97/3 against beam exposure. Therefore, we are confident that the FE-\nAFE transition captured in our in-situ TEM measurements is triggered by the\ntemperature variation rather than by electron-beam effects. Meanwhile, as shown in the\ndielectric temperature spectrum in Fig. S3, a phase transition occurred at approximately\n-20 ℃. Even in electrical performance tests, phase transitions may occur even if the\nirradiation effects of electron beams and neutron sources are not involved. In summary,\nthe observed phase transition reflects the intrinsic response of PZT97/3 to temperature\nchanges, independent of any extrinsic influences from the imaging conditions.\nFig. R12| Representative TEM frames extracted from the continuous recording of\nPZT97/3 after cooling to -50 ℃. Shown are the initial state (0 min), and frames at\napproximately 30 min, 1 h, and 1.5 h.\nFig. R13| TEM images and corresponding SAED patterns of the PZT97/3 before\nand after prolonged electron-beam irradiation at room temperature. Low-\nmagnification TEM images (left) and SAED patterns (right) of the PZT97/3 taken along\nthe [001] zone axis before (a) and after (b) continuous e-beam irradiation for\np\napproximately two hours. The scale bars in TEM and SAED images correspond to 100\nnm and 2 nm-¹, respectively.\n3. The plots in Fig. 4 and Fig. S7 imply that at high temperatures, the R3c phase is\npreferred and the R3c to Pbam energy barrier is low. Theoretically, what would be the\ntemperature at which this phase transition is likely to happen within the window of the\nmeasurement?\nResponse: Thanks for your comments. We apologize for any misunderstanding our\nprevious statement may have caused. According to our theoretical calculations (Figs. 4\nand S7), the actual situation is as follows: at low temperatures, the energy barrier for\nthe transition from the R3c phase to the Pbam phase is low, while at room temperatures,\nthe energy barrier for the reverse transition from the Pbam phase back to the R3c phase\nis high. This key distinction implies that overcoming this high barrier through thermal\nagitation alone to achieve the reverse phase transition is extremely difficult. During our\nTEM experiments, when the temperature was increased from low to high, we did not\nobserve the phase transition from Pbam back to R3c. We further heated the sample to\n100 °C and maintained it for a period (Fig. S4), yet no signs of a Pbam-R3c transition\nwere detected (Fig. R15). In conclusion, we believe that the simple heating process\nalone is insufficient to cause the PZT97/3 material to undergo the Pbam-R3c phase\ntransformation.\na. It seems from the theory that this phase transition is not irreversible, but hysteretic.\nThe data would be strengthened greatly by observing the AFE-FE phase transition, and\nreporting the temperature of it. Is my understanding correct, and is this a possible\nmeasurement?\nResponse: We sincerely thank the reviewer for the professional comments.\nYour understanding is theoretically reasonable, as the presence of an energy barrier typically\nindicates possible hysteresis. However, as show in Fig. R15, our high-temperature\nexperiments did not observe the reverse AFE-FE transition, suggesting that the\ntransition is effectively irreversible under thermal driving alone. We therefore believe\nthat observing the AFE-FE reverse transition within this thermal window is not feasible\nunder the current temperature-only conditions.\nb. It would be interesting to compare a theoretically derived hysteresis for the phase of\nthe material and compare it to an experimentally verified hysteresis, and would greatly\nstrengthen the results of the paper.\nResponse: Thanks for your comments. Your suggestion has clearly outlined the\ndirection for our next steps. We plan to apply additional driving forces, such as electric\nfields or stress, to attempt to trigger and investigate the hysteresis during the phase\ntransition process. Once again, we sincerely thank you for your insightful comments,\nwhich have greatly enhanced the depth of our response and the overall quality of the\npaper. We hope that the above clarifications and explanations have adequately\naddressed your questions.\nFig. R14| (a) The calculated pathway for phase transition from AFE to FE at -\n50 ℃. (b) The calculated pathway for phase transition from FE to AFE at 25 ℃.\nFig. R15| Temperature-dependent TEM images of an antiferroelectric material\nacquired at (a) -170 ℃, (b) -50 ℃, (c) 0 ℃, (d) 25 ℃, and (e) 100 ℃.\n4. Line 272-273 states that the intervention of external energy is necessary for the AFE-\nFE transition. Is this done to the samples in question, and confirmed? It would\nstrengthen the results to see this done. For example, could Fig. S3 be related to this\nclaim?\nResponse: As shown in Fig. R16(a), the phase transition path of PZT97/3 is illustrated.\nPZT97/3 remains in the FE (R3c) phase at room temperature. During cooling, neutron\ndiffraction patterns of PZT97/3 (Fig. R16-b) show the emergence of new superlattice\ndiffraction peaks at approximately -50 ℃, indicating the occurrence of the FE (R3c)-\nAFE (Pbam) phase transition. After the FE-AFE transition is completed, the sample\nwas heated to 25 ℃, yet no phase transition occurred, indicating that PZT97/3 cannot\nundergo the AFE-FE transition without external energy input. As shown in Fig. R16(c),\nwhen an external electric field was applied to the PZT97/3 sample at 25 ℃, the\nsuperlattice diffraction peaks disappeared in the XRD pattern, confirming the\noccurrence of the AFE-FE transition. Meanwhile, the dielectric temperature spectra\n(Fig. S3) display a mutation near -20 ℃ during heating under an applied field. This\nmutation may occur due to a phase change, but it can only serve as an auxiliary for the\nXRD data and cannot independently prove that a phase change has taken place.\nTherefore, the AFE-FE transition of PZT97/3 necessarily requires the introduction of\nexternal energy. These results, together with previous reports that external electric fields\ncan drive the AFE-FE transition in PZT ceramics at room temperature [7-10],\nconsistently demonstrate that the AFE-FE transition in PZT97/3 instead requires\nexternal energy input to overcome the transition barrier. This further confirms that\nexternal energy input is indispensable for enabling the AFE-FE transition at room\ntemperature.\nFig. R16| The phase structure evolution mechanism of PZT97/3. (a) The phase\ntransition path of PZT97/3. (b) The neutron diffraction patterns of the PZT97/3\nceramics at 25 ℃, -50 ℃, and 25 ℃, respectively. (c) The XRD of the PZT97/3\nceramics at 25 ℃, -50 ℃, and 25 ℃, respectively.\nReferences:\n[7] Y. L. Wang, Z. M. Cheng, Y. Sun, X. Dai. Phase transition study of PZT 95/5 ceramics. Phys. Rev. E\n150 168-174 (1998).\n[8] V. M. Ishchuk. The Problem of Phase Transitions in PZT Based Solid Solutions, Pb(Mg Nb )O ,\n1/3 2/3 3\nPb(In Nb )O and \"Relaxor Ferroelectricity.\" 1. Experimental Results and Phase Diagrams for PZT\n1/2 1/2 3\nBased Solid Solutions. Ferroelectr. 289 1-32 (2003).\n[9] Q. N, Zheng, T. Q. Yang, K. Wei, J. F. Wang, X. Yao. Effect of Sn:Ti variations on electric filed\ninduced AFE-FE phase transition in PLZST antiferroelectric ceramics. J. Ceram. Int. 38 S9-S12 (2012).\n[10] R. N. Singh. Large Strain Ceramics Based on Electric Field-Induced Phase Transformation in Sr-\nModified PZT. Ferroelectr. 293 219-229 (2003).\nAction: We have added following discussions to the revised manuscript. (Please see\nthe parts marked by red color on page 11 of the revised manuscript)\nAs show in Fig. S8, after the FE-AFE transition is completed upon cooling, reheating\nto room temperature does not trigger the reverse transformation under zero-field\nconditions, suggesting that thermal energy alone is insufficient to overcome the\ntransition barrier. In contrast, when an external electric field is applied at room\ntemperature, the AFE superlattice reflections disappear, indicating that the AFE-FE\ntransition can occur under the applied field. This behavior is consistent with previous\nobservations in PZT-based ceramics [41-44]. The intervention of other external field\nenergy, such as electric field or irradiation field, is necessary for the AFE-FE transition.\nAction: We have added following figure to the revised supplemental material. (Please\nsee the parts marked by red color on page 9 of the revised supplemental material)\nFig. S8. The phase structure evolution mechanism of PZT97/3. (a) The phase transition\npath of PZT97/3. (b) The neutron diffraction patterns of the PZT97/3 ceramics at 25 ℃,\n-50 ℃, and 25 ℃, respectively. (c) The XRD of the PZT97/3 ceramics at 25 ℃, -50 ℃,\nand 25 ℃, respectively.\n5. From line 284-285: how do the Gibbs free energy and the DFT calculations get\nincorporated into the phase field simulations? It is not clear from the methods section\nor main text.\nResponse: We sincerely thank the reviewer for this insightful question. Through first-\nprinciples DFT calculations, we obtained the temperature-dependent free energy\nfunctions of the AFE and FE phases, as shown in Fig. S7. In our phase-field model,\nthese DFT-derived free energies are incorporated in the form of a Landau free energy,\nfollowing the approach of Ref. [11, 12]. To establish a quantitative link between the\nfirst-principles results and the phase-field model, we parameterized the DFT-computed\nfree energies of the FE and AFE phases using a sixth-order Landau expansion[13]：\nlandau\n2 4 6 1 2 2 4 2\n𝑓 = ∑𝑖(𝛼1𝑃𝐹𝑖 +𝛼11𝑃𝐹𝑖 +𝛼111𝑃𝐹𝑖)+2𝛼12∑𝑖≠𝑗𝑃𝐹𝑖𝑃𝐹𝑗 +𝛼112∑𝑖≠𝑗𝑃𝐹𝑖𝑃𝐹𝑗 +\n2 2 4 6 1 2 2 4 2\n𝛼123∏𝑖𝑃𝐹𝑖 +∑𝑖(𝛽1𝑃𝐴𝑖 +𝛽11𝑃𝐴𝑖 +𝛽111𝑃𝐴𝑖)+2𝛽12∑𝑖≠𝑗𝑃𝐴𝑖𝑃𝐴𝑗 +𝛽112∑𝑖≠𝑗𝑃𝐴𝑖𝑃𝐴𝑗 +\n(\n1\n)\n𝛽H 1 e 2 r 3 e,∏ 𝑖𝑃𝐴 𝑖a+nd𝜇 11∑ 𝑖 d𝑃e𝐹n𝑖𝑃o𝐴te𝑖 +th𝜇e 1 p 2 o∑la 𝑖≠ ri 𝑗 z𝑃at𝐹i𝑖o𝑃n𝐴 𝑗c+om2p𝜇o 4 n 4 e∑nt 𝑖 s ≠ 𝑗 o𝑃f 𝐹 t 𝑖 h𝑃e 𝐹 𝑗 F𝑃𝐴 E 𝑖 𝑃a 𝐴 n 𝑗 d A F E p h ases,\nrespect\n𝑃\niv𝐹e𝑖ly. Th\n𝑃\ne𝐴 𝑖temperature-dependent coefficients and are modeled as\nCurie-Weiss functions [13]: 𝛼1 𝛽1\n(\n2\n)\nwhere T is the temperatu𝛼re 1 ,( 𝑇) =is the vac,u𝛽u 1 m(𝑇 p)er=mittivity, and C is the Curie constant.\n2𝜀0𝐶 2𝜀0𝐶\nand represent t\n𝜀\nh0e Curie-Weiss temperatures of the ferroelectric and\n𝑇\nan0tiferroel\n𝑇\ne0c𝐴t𝐹ri𝐸c phases. The higher-order coefficients are treated as temperature-\nindependent constants. For the FE（ , ）and\n𝑃𝐹1 = 𝑃𝐹2 = 𝑃𝐹3 = 𝑃𝐹 𝑃𝐴1 = 𝑃𝐴2 = 𝑃𝐴3 = 0\nAFE（ , ）configurations, the Landau free\nenergie𝑃s 𝐹 c 1 a=n b𝑃e 𝐹 2 ex=pr𝑃e 𝐹 s 3 se=d i0nd𝑃i 𝐴 v 1 id=ua𝑃ll 𝐴 y 2 a=s: 𝑃𝐴,𝑃𝐴3 = 0\n, (1)\nlandau_FE\n2 4 6\n𝑓 = 3𝛼1(𝑇)𝑃𝐹 +3(𝛼11 +𝛼12)𝑃𝐹 +(3𝛼111 +𝛼112 +𝛼123)𝑃𝐹\n. (2)\nlandau_AFE\n2 4 6\nBecause th𝑓e DFT ca=lcu2la𝛽t 1 io(𝑇ns) 𝑃d 𝐴 o n+ot( 2d𝛽ir 1 e 1 ct+ly𝛽 y 1 i 2 e)l𝑃d 𝐴 th+e p2o(l𝛽a 1 r 1 iz 1 a+tio𝛽n 1 1 v 2 a)l𝑃u 𝐴 es of the FE\nand AFE phases at different temperatures, polarization is treated as an implicit variable\nthat can be determined indirectly through the equilibrium condition:\n, (5)\n𝜕𝑓𝑙𝑎𝑛𝑑𝑎𝑢_𝐹𝐸 3 5\n𝜕𝑃𝐹 =0⇒6α1(T)PF+12(α11+α12)PF +6(3α111+α112+α123)PF =0\n(6)\n𝜕𝑓𝑙𝑎𝑛𝑑𝑎𝑢_𝐹𝐸 3 5\nBy numerically 𝜕s𝑃o𝐹lving⇒ thβ4es 1 eT( twP)o A e+q4u(aβ2tion 11 s+,β th 1 e 2 )pPA olar+iz1a2tβi(on 1 1 v 1 a+βlue 1 s 1 2PP)A T=,0P. T at\nF A\neach temperature are obtained. Subsequently, the following fitting functions are\n( ) ( )\nconstructed:\n, (3)\n𝑅𝐹𝐸 = 𝑓𝐷𝐹𝑇_𝐹𝐸(𝑇)−𝑓𝑙𝑎𝑛𝑑𝑎𝑢_𝐹𝐸(𝑃𝐹(𝑇),𝑇) . (4)\n𝑅𝐴𝐹𝐸 = 𝑓𝐷𝐹𝑇_𝐴𝐹𝐸(𝑇)−𝑓𝑙𝑎𝑛𝑑𝑎𝑢_𝐴𝐹𝐸(𝑃𝐴(𝑇),𝑇)\nThe residuals of Eqs. (7)-(8) are minimized via a nonlinear least-squares fitting over all\ntemperature points. The optimized Landau parameters successfully reproduce the DFT-\ncalculated energy hierarchy and the FE-AFE transition behavior across the entire\ntemperature range. This parameterization process establishes a quantitative bridge\nbetween first-principles calculations and the phase-field model, ensuring that the\nsimulated free-energy landscape accurately reflects the DFT-derived energetics. Fig.\nR17 shows the DFT free energy data points and the corresponding Landau fitting curves.\nIt is worth noting that other parameters in the phase-field model (such as elastic\nconstants, electrostrictive coefficients, and gradient coefficients) were taken from Ref.\n[14], which are fitted to experimental phase diagrams.\nFig. R17| Comparison of DFT-calculated and phase-field-fitted free energies over\nthe temperature range, showing the agreement between first-principles\ncalculations and model predictions.\nReferences:\n[11] J. Íñiguez, S. Ivantchev, J. M. Perez-Mato, A. García. Devonshire-Landau free energy of BaTiO\n3\nfrom first principles. Phys. Rev. B 63 144103 (2001).\n[12] D. Durdiev, F. Wendler, M. Zaiser, H. Azuma, T. Tsuzuki, S. Ogata, M. Uranagase. Parameterization\nof a phase field model for ferroelectrics from molecular dynamics data. Acta Mater. 283 120513 (2025).\n[13] M. J. Haun, T. J. Harvin, M. T. Lanagan, Z. Q. Zhuang, S. J. Jang, L. E. Cross. Thermodynamic\ntheory of PbZrO . J. Appl. Phys. 65 3173-3180 (1989).\n3\n[14] Y. Peng, Y. Jiang, X. Wang. Effect of oxygen octahedral tilt on ferroelectric-antiferroelectric\ntransition of PZT95/5 induced by hydrostatic pressure and electric field: A phenomenological\nthermodynamic analysis. J. Eur. Ceram. Soc. 44 5794-5803 (2024).\nAction: We have added following discussions to the revised manuscript. (Please see\nthe parts marked by red color on page 12 of the revised manuscript)\nTo describe the dynamic process of the FE-AFE phase transition with high fidelity, a\nmultiscale theoretical framework was established. We conducted two-dimensional (2D)\nphase-field simulations to investigate the FE-AFE phase transition at low temperatures,\nwith the Gibbs free energy (ΔG) directly derived from density functional theory (DFT)\ncalculations. Specifically, the DFT-calculated Gibbs free energies of the FE and AFE\nphases were parameterized using a sixth-order Landau expansion, in which the\ntemperature-dependent coefficients were modeled as Curie-Weiss functions and the\nhigher-order terms were treated as temperature-independent constants. The resulting\nLandau coefficients quantitatively reproduce the DFT-computed energy hierarchy\nacross the full temperature range. These parameterized free-energy functions were then\nincorporated into the phase-field simulations as the thermodynamic driving force,\nensuring a direct and physically consistent link between first-principles energetics and\nmesoscale evolution.\n6. In general, I find that the phase field simulation work is not well-explained and not\nput into context. The results are very interesting: namely figure 5e, but I do not\nunderstand how the data in figure 5e comes from the phase field analysis in the methods\nsection. What parameters are fitted for? Which parameters are simulated? Which\nparameters, in the long description of the phase field method and equations, are the\nmost important for estimating the domain wall velocity and the temperature dependency?\na. More time should be spent describing figure 5e: how does the phase field prediction\ndo such a good job of modeling the system? Unless I missed something, the only\nempirical parameters used in the simulations were the lattice sizes of the crystals in\nquestion…\nResponse: We sincerely thank the reviewer for this thoughtful comment and for\nrecognizing the value of the phase-field results. We agree that additional clarification is\nneeded to better connect the phase-field formulation with the data presented in Fig. 5e.\nThe following explanation has been added to the revised manuscript.\nIn our phase-field simulations, all quantities were nondimensionalized. The temporal\nand spatial evolution of the ferroelectric (FE) and antiferroelectric (AFE) order\nparameters was obtained by solving the time-dependent Ginzburg-Landau (TDGL)\nequation:\n, (5)\n𝜕𝑃𝐹𝑖 𝛿𝐹\n𝜕𝑡 = −𝐿𝐹𝐸 𝛿𝑃𝐹𝑖\n. (6)\n𝜕𝑃𝐴𝑖 𝛿𝐹\nwhere L and is the kineti𝜕c𝑡 co=eff−ic𝐿i 𝐴 e 𝐹 n 𝐸 t f𝛿o𝑃r𝐴 𝑖the FE and AFE order parameters,\nFE\nand F is the total f\n𝐿\nrAeeFE energy functional. The migration velocity of the AFE-FE interface\nwas calculated under different temperature conditions, where the computed velocities\nare also in dimensionless form, which were directly obtained from the phase-field\nsimulations described in the Methods section in Fig. 5e.\nTo compare the simulated results with experimental measurements, a dimensional\ntransformation was performed by introducing appropriate physical length l and time\n0\nτ scales. For the Allen-Cahn-type non-conserved TDGL model, the physical time\ns0cale τ is related to the kinetic coefficient L through the following expression [15,\n16]: 0\n, (7)\n1\nwhere is the free energy difference𝜏 b 0 e=tw𝐿e𝛥e𝑓n the FE and AFE phases. Consequently,\nthe physical domain-wall velocity can be expressed as:\n𝛥𝑓\n𝑣𝑝ℎ𝑦𝑠 , (8)\n𝑙0\nwhere is the dimensio𝑣n 𝑝 l ℎ e 𝑦 s 𝑠 s =ve𝑣lo 𝑠𝑖 c 𝑚 ity𝜏0 o=bt𝑣ai 𝑠 n 𝑖𝑚 ed𝑙0 f𝐿r𝛥om𝑓 the phase-field simulation,\nis d\n𝑣\ne𝑠r𝑖i𝑚ved from DFT-calculated FE-AFE energy differences, and τ are the\n𝛥 ch 𝑓 aracteristic length and time scales fitted to match experimental da 𝑙 ta0. 0\nIn evaluating the domain-wall velocity and its temperature dependence, among the\nvarious parameters in the phase-field model, the gradient energy coefficient GF, GA\ndetermines the domain-wall width and energy, while the kinetic coefficient L , L\n(FE AFE)\n( )\ngoverns the dynamic response of domain-wall motion. In the Landau free energy, the\nsecond-order coefficient , which follows the Curie-Weiss law, are the key\ntemperature-dependent par\n(\n\na\n𝛼\nm1, e t\n𝛽\ne1r\n)\ns:\n, (9)\n𝑇−𝑇0 𝑇−𝑇0𝐴𝐹𝐸\nwhere is the temperature𝛼, 1(𝑇 )is= th2e𝜀 0v𝐶a,c𝛽u 1 u(m𝑇 )pe=rmi2tt𝜀i0v𝐶ity, and C is the Curie constant.\nand\n𝑇\nrepresent the\n𝜀\nC0urie-Weiss temperatures of the FE and AFE phases.\n𝑇0 𝑇 d0𝐴ic𝐹t𝐸ate the temperature dependence of the free energy difference between the\n(\nF\n𝛼\nE1\n,\na n\nd\n𝛽\n1\n)\nAFE phases-i.e., the thermodynamic driving force for phase transformation.\nThese parameters therefore crucial for determining the domain wall velocity and its\ntemperature dependence.\nThe good agreement between the simulated and experimental results in Fig. 5e arises\nfrom the self-consistent coupling of the polarization, elastic, and thermal fields in our\nmodel. In our simulations, all material parameters-including the Landau coefficients,\nelastic constants, electrostrictive coefficients, and gradient energy coefficients-were\nobtained directly from first-principles calculations or experimental data reported in the\nliterature [17-19]. The kinetic coefficient , which plays the most critical\nrole in governing the phase transition\n(\nr\n𝐿\na𝐹te𝐸,\n, 𝐿\nw𝐴a𝐹s𝐸\n)\ndetermined by fitting to the\nexperimentally measured domain-wall velocity. Therefore, the accurate reproduction of\nthe domain wall velocity and its temperature dependence in Fig. 5e is a physically\nconsistent outcome of the multiscale model, achieved by introducing a realistic,\nexperimentally validated temperature-dependent mobility rather than empirical\noverfitting.\nReferences:\n[15] A. Karma, W. J. Rappel. Quantitative phase-field modeling of dendritic growth in two and three\ndimensions. Phys. Rev. E 57 4323-4349 (1998).\n[16] L. Q. Chen. Phase-field models for microstructure evolution. Title Annu. Rev. Mater. Res. 32 113-\n140 (2002).\n[17] M. J. Haun, T, J, Harvin, M, T, Lanagan, Z. Q. Zhuang, S. J. Jang, L. E. Cross. Thermodynamic\ntheory of PbZrO . J. Appl. Phys. 65 3173-3180 (1989).\n3\n[18] Z. Fan, F. Xue, G. Tutuncu, L. Q. Chen, X. Tan. Interaction Dynamics Between Ferroelectric and\nAntiferroelectric Domains in a PbZrO -Based Ceramic. Phys. Rev. Appl. 11 064050 (2019).\n3\n[19] Y. Peng, Y. Jiang, X. Wang. Effect of oxygen octahedral tilt on ferroelectric-antiferroelectric\ntransition of PZT95/5 induced by hydrostatic pressure and electric field: A phenomenological\nthermodynamic analysis. J. Eur. Ceram. Soc. 44 5794-5803 (2024).\nAction: We have added following discussions to the revised manuscript. (Please see\nthe parts marked by red color on page 13 of the revised manuscript)\nThe resulting Landau coefficients quantitatively reproduce the DFT-computed energy\nhierarchy across the full temperature range. To ensure a physically consistent\nconnection between atomic-scale energetics and mesoscale evolution, the\nparameterized Landau free-energy functions were incorporated into the phase-field\nsimulations as the thermodynamic driving force, while the elastic, electro strictive, and\ngradient energy coefficients were taken from experimental or literature values. Among\nthese parameters, the gradient energy coefficient controls the domain-wall width and\nenergy, whereas the kinetic coefficient (L) governs the dynamic response of domain-\nwall motion. The second-order Landau coefficient, which follows the Curie-Weiss law,\ndetermines the temperature dependence of the FE-AFE free-energy difference-thus\nproviding the key thermodynamic driving force for the phase transition. The kinetic\ncoefficient was the only adjustable parameter, fitted to the experimentally measured\ndomain-wall velocity to establish the time scale of the simulations. Using this\nmultiscale model, we performed simulations of the FE-AFE phase transition under low-\ntemperature conditions. Below the critical transition temperature, a local structural\nperturbation-such as a microcrack, inclusion, or dislocation-was introduced to initiate\nnucleation. As illustrated in Fig. 5a, the antiferroelectric phase preferentially nucleates\nat the perturbation site and subsequently propagates outward, forming a curved domain\nwall that advances with a diffusion-like velocity V. This process progressively\ntransforms the ferroelectric phase into the antiferroelectric phase over time (Video S2).\nFig. 5e presents the computed domain-wall velocity as a function of temperature,\ndirectly obtained from the nondimensionalized phase-field simulations and then\nconverted to physical units through the kinetic coefficient described above. The results\nshow that the phase-transition speed increases significantly as temperature decreases.\nThe velocity follows a near-exponential dependence on temperature, eventually\nsaturating at a critical value (V =8.05 nm/s). This behavior arises from the reduction\nlimit\nin the lattice transformation barrier at lower temperatures (Fig. S7), which facilitates\neasier structural reconfiguration along the transformation pathway. The excellent\nagreement between the simulated and experimental data demonstrates that the self-\nconsistent coupling of polarization, elastic, and thermal fields in the model successfully\ncaptures the intrinsic temperature-dependent kinetics of the FE-AFE interface\nmigration.\n7. Line 298 - 299: “This behavior [of the velocity saturating at 8.05 nm/s] is attributed\nto the reduction in the lattice transformation barrier at lower temperatures.” How is\nthis conclusion drawn, and what part of the phase field simulations or equations lead\nus to that conclusion?\nResponse: We sincerely thank the reviewer for this insightful question. We agree that\nthe explanation in the original manuscript was too brief, and we have now expanded\nthis section to clarify how this conclusion is derived from the phase-field simulations.\nIn the phase-field model, the domain-wall velocity V is governed by the balance\nbetween the thermodynamic driving force-represented by the free-energy difference\nbetween the FE and AFE phases-and the kinetic coefficient [20, 21]:\n. (10)\n𝛥𝑓 𝐿(𝑇)\nThis relation is derived from the non-conserved time-dependent Ginzburg-Landau\n𝑉 ∝ 𝐿(𝑇)⋅𝛥𝑓\n(TDGL) equation under steady-state conditions. As shown in Fig. S7, the phase-field\nsimulations reveal that the free-energy barrier associated with the lattice transformation\ngradually decreases as temperature decreases, leading to a larger and therefore a\nstronger thermodynamic driving force. This explains the initial rapid increase in the\n𝛥𝑓\ntransformation velocity at low temperatures (as observed in Fig. 5e). However, once\nthe temperature drops below a critical threshold, the energy barrier becomes nearly\neliminated, and the system approaches its lowest-energy configuration. In this regime,\nthe phase-transition kinetics are no longer limited by the thermodynamic driving force\nbut by intrinsic dynamic factors, such as structural rearrangement and polarization\nrelaxation [20, 21]. Consequently, the migration velocity becomes constrained by the\nkinetic coefficient , which tends to saturate or vary weakly at low temperatures. As a\nresult, the overall domain-wall velocity V reaches a limiting value (V =8.05 nm/s),\n𝐿 limit\nreflecting the intrinsic dynamical limit of the lattice transformation process rather than\nthermodynamic control.\nReferences:\n[20] S. M. Allen, J. W. Cahn. A microscopic theory for antiphase boundary motion and its application to\nantiphase domain coarsening. Acta Metall. 27 1085-1095 (1979).\n[21] L. Q. Chen, J. Shen. Applications of semi-implicit Fourier-spectral method to phase field equations.\nComput. Phys. Commun. 108 147-158 (1998).\nAction: We have added following discussions to the revised manuscript. (Please see\nthe parts marked by red color on page 13 of the revised manuscript)\nAmong these parameters, the gradient energy coefficient controls the domain-wall\nwidth and energy, whereas the kinetic coefficient (L) governs the dynamic response of\ndomain-wall motion. In the phase-field framework, the domain-wall velocity V follows\na linear relationship with the local free-energy difference and the kinetic\ncoefficient, expressed as [45, 46], where represents the\n𝛥𝑓\ntemperature-dependent interfacial mobility. The second-order Landau coefficient,\n𝑉 ∝ 𝐿(𝑇)⋅𝛥𝑓 𝐿(𝑇)\nwhich follows the Curie-Weiss law, determines the temperature dependence of the FE-\nAFE free-energy difference-thus providing the key thermodynamic driving force for\nthe phase transition. The kinetic coefficient was the only adjustable parameter, fitted to\nthe experimentally measured domain-wall velocity to establish the time scale of the\nsimulations. Using this multiscale model, we performed simulations of the FE-AFE\nphase transition under low-temperature conditions. Below the critical transition\ntemperature, a local structural perturbation-such as a microcrack, inclusion, or\ndislocation-was introduced to initiate nucleation. As illustrated in Fig. 5a, the\nantiferroelectric phase preferentially nucleates at the perturbation site and subsequently\npropagates outward, forming a curved domain wall that advances with a diffusion-like\nvelocity V. This process progressively transforms the ferroelectric phase into the\nantiferroelectric phase over time (Video S2). Fig. 5e presents the computed domain-\nwall velocity as a function of temperature, directly obtained from the\nnondimensionalized phase-field simulations and then converted to physical units\nthrough the kinetic coefficient described above.\nAs shown in Fig. S7, the simulations reveal that the lattice-transformation energy\nbarrier gradually decreases as temperature decreases, leading to a larger Δf and a\ncorrespondingly stronger thermodynamic driving force. This explains the initial rapid\nincrease of the phase-transition velocity at low temperatures (Fig. 5e). However, once\nthe temperature falls below a critical threshold, the energy barrier becomes nearly\neliminated, and the system approaches its minimum-energy configuration. In this\nregime, the domain-wall motion is no longer limited by the thermodynamic driving\nforce but by intrinsic kinetic factors-such as structural rearrangement and polarization\nrelaxation [45, 46]-which cause the kinetic coefficient to saturate. Consequently,\nthe overall domain-wall velocity V reaches a limiting value (V =8.05 nm/s ),\n𝐿(𝑇) limit\nrepresenting the intrinsic dynamical limit of the lattice-transformation process rather\nthan thermodynamic control. The excellent agreement between the simulated and\nexperimental data demonstrates that the self-consistent coupling of polarization, elastic,\nand thermal fields in the model successfully captures the intrinsic temperature-\ndependent kinetics of the FE-AFE interface migration.\n8. In general, the phase-field section of the methods section is very poorly connected to\nthe main text. There is a lot of math and work that goes into it, but there is little\nindication of how that math comes together to create Figure 5. For example,\nResponse: We sincerely thank the reviewer for this valuable comment. We agree that\nthe connection between the mathematical formulation of the phase-field model and its\nphysical interpretation in the main text needs to be clarified.\na. What drives the transition?\nResponse: In our simulations, the driving force for the FE-AFE phase transition arises\nfrom the local free-energy difference ( ) between the two phases at a given\ntemperature [22]. Variations in temperature modify the shape and depth of the Landau\n𝛥𝑓\ndouble-well potential, leading to an energy imbalance between the FE and AFE states.\nThis imbalance provides the thermodynamic driving force that induces phase\ntransformation and promotes the motion of the phase boundary. Mathematically, this\ndriving force appears as the functional derivative of the total free energy with respect\nto the polarization order parameter, i.e., , in the time-dependent Ginzburg–Landau\n𝛿𝐹\n(TDGL) equation. Physically, it represen𝛿ts𝑃 𝐹th/𝐴e tendency of the system to evolve through\ndomain reconfiguration in order to minimize its total free energy.\nb. What parameters drive the diffusion of the domain wall?\nResponse: Within the phase-field framework, the diffusion of the domain wall is\ngoverned by both the thermodynamic driving force and the kinetic mobility. The\nthermodynamic driving force originates from the local free-energy difference ( )\nbetween the FE and AFE phases, which provides the impetus for the interface to move\n𝛥𝑓\nin the direction that reduces the total free energy [23]. The key kinetic parameter\ncontrolling the rate of this motion is the kinetic coefficient in the time-dependent\nGinzburg-Landau (TDGL) equation, which determines how quickly the polarization\n𝐿\nfield evolves over time. The temperature dependence of captures the thermally\nactivated nature of domain-wall diffusion. In addition, the gradient energy coefficient\n𝐿(𝑇)\nand elastic coupling coefficients influence the diffusion kinetics by\nmodifying the domain-wall width, interfacial energy, and strain compatibility [24].\n(𝐺𝐹,𝐺𝐴)\nOverall, the diffusion rate of the domain wall results from the interplay between the\nthermodynamic driving force ( ) and the kinetic coefficient , modulated by the\ninterfacial and elastic energy terms. These parameters collectively determine the\n𝛥𝑓 𝐿(𝑇)\ntemperature-dependent diffusion behavior observed in Fig. 5.\nc. How is the microstructure defect implemented?\nResponse: In the numerical implementation, microstructural inhomogeneities were\nintroduced as localized perturbations of the order parameter field within the simulation\ndomain [23, 24]. Specifically, several grid points were randomly selected and assigned\nsmall fluctuations (within ±5%) in the local order parameter. This approach mimics the\nlocal compositional or structural irregularities commonly observed in real ferroelectric-\nantiferroelectric systems, such as dislocations, point defects, or compositional\ninhomogeneities. The perturbation amplitude was carefully limited to less than 5% to\nmaintain physical realism and numerical stability.\nMathematically, these local perturbations modify the free-energy landscape by\nintroducing small spatial variations in the total free-energy functional F. Such\nvariations locally alter the thermodynamic driving force , thereby acting as\n𝛿𝐹\npinning centers or nucleation sites in the evolution governe𝛿d𝑃 𝐹b/y𝐴 the time-dependent\nGinzburg-Landau (TDGL) equation.\nPhysically, the defect-induced energy fluctuations lead to nonuniform domain-wall\nmotion, interface bending, and localized pinning-depinning dynamics, consistent with\nthe microstructural behavior observed in the simulations (Fig. 5). This treatment allows\nthe model to reproduce the experimentally observed microstructural features, including\ncurved domain walls and heterogeneous phase evolution, thus providing a realistic\nrepresentation of defect-domain interactions in the FE-AFE transition process.\nd. What parameters are responsible for the temperature dependency of the velocity?\nResponse: The temperature dependence of the domain-wall velocity in our phase-field\nsimulations originates from the explicit temperature dependence of two key parameters\nin the Landau-Ginzburg-Devonshire (LGD) free-energy formulation [25]: the Landau\ncoefficient α T ,β T and the kinetic coefficient L . The Landau coefficient\n1 1\ngoverns the [cur(va) ture (of )t]he double-well potential and thu(s𝑇 d)etermines the free-energy\ndifference between the ferroelectric (FE) and antiferroelectric (AFE) phases. As\nthe temperature decreases, the potential wells deepen and the energy barrier between\n𝛥𝑓\nthe two phases changes accordingly, directly affecting the thermodynamic driving force\nfor domain-wall motion [26]. Meanwhile, the kinetic coefficient L controls the rate\nat which the polarization field evolves in response to this driving force. Its temperature\n(𝑇)\ndependence reflects the intrinsic relaxation dynamics of lattice and polarization\nrearrangement. Therefore, the combined effects of α T ,β T -which modulates the\n1 1\nthermodynamic driving force-and L -which go[ve(rns) the( k)i]netic response-jointly\ndetermine the temperature-dependent evolution of the domain-wall velocity observed\n(𝑇)\nin Fig. 5e.\ne. Where does the DFT work fit into the phase-field simulations?\nResponse: In our work, the DFT calculations serve as the atomic-scale foundation for\nparameterizing the Landau-Ginzburg-Devonshire (LGD) free-energy potential used in\nthe phase-field simulations. Specifically, the DFT-calculated total energies as a function\nof temperature were fitted using a sixth-order Landau polynomial expansion, where the\ncoefficient were determined to reproduce the DFT-computed energy hierarchy of the\nferroelectric (FE) and antiferroelectric (AFE) phases. The temperature dependence of\nthe second-order coefficient capture the thermodynamic evolution of\nthe system with temperature. T\n𝛼\nh1i\n(\n𝑇\n),\n𝛽\n1\n(\n𝑇\n)\nhis DFT-based parameterization ensures that the phase-\nfield simulations inherit a quantitatively accurate and physically consistent free-energy\nlandscape derived from first principles. In other words, the DFT results provides the\nmicroscopic energetic input, while the phase-field model describes the mesoscale\nkinetics and domain evolution based on that input, thus achieving a multiscale coupling\nbetween atomic-scale energetics and continuum-scale dynamics.\nThis procedure, as described in detail in our response to Comment (5), quantitatively\nlinks the DFT-derived energetics to the mesoscale phase-field simulations, ensuring a\nconsistent multiscale description of the FE-AFE transition.\nReferences:\n[22] S. M. Allen, J. W. Cahn. A microscopic theory for antiphase boundary motion and its application to\nantiphase domain coarsening. Acta Metall. 27 1085-1095 (1979).\n[23] L.Q. Chen. Phase-field models for microstructure evolution. Title Annu. Rev. Mater. Res. 32 113-\n140 (2002).\n[24] Y. Wang, A. G. Khachaturyan. Three-dimensional field model and computer modeling of martensitic\ntransformations. Acta Metall. 45 759-773 (1997).\n[25] A. F. Devonshire. XCVI. Theory of barium titanate. Philos. Mag. J. Sci. 40 1040–1063 (1949).\n[26] J. Wang, S. Q. Shi, L. Q. Chen, Y. Li, T. Y. Zhang. Phase-field simulations of\nferroelectric/ferroelastic polarization switching. Acta Metall. 52 749-764 (2004).\nAction: We have added following discussions to the revised manuscript. (Please see\nthe parts marked by red color on page 12 of the revised manuscript)\nTo describe the dynamic process of the FE-AFE phase transition with high fidelity, a\nmultiscale theoretical framework combining first-principles energetics and mesoscale\nkinetics was established. We conducted two-dimensional (2D) phase-field simulations\nto investigate the FE-AFE phase transition at low temperatures, with the free energy\n(ΔG) directly derived from density functional theory (DFT) calculations. In this\nframework, the DFT calculations provide the atomic-scale foundation for constructing\nthe Landau-Ginzburg-Devonshire (LGD) free-energy functional. Specifically, the DFT-\ncalculated total energies of the FE and AFE phases were fitted by a sixth-order Landau\npolynomial expansion, where the coefficients were optimized to reproduce the DFT-\ncomputed energy hierarchy and its temperature evolution. The temperature dependence\nof the second-order Landau coefficient follows the Curie-Weiss law, capturing the\nthermodynamic evolution of the free-energy difference between the FE and AFE phases.\nThis DFT-based parameterization ensures that the phase-field simulations inherit a\nquantitatively accurate free-energy landscape derived from first principles, thereby\nbridging the atomic and mesoscale descriptions.\nTo ensure a physically consistent connection between atomic-scale energetics and\nmesoscale evolution, the parameterized Landau free-energy functions were\nincorporated into the phase-field simulations as the thermodynamic driving force for\nphase transformation, while the elastic, electrostrictive, and gradient energy\ncoefficients were taken from experimental or literature values. The thermodynamic\ndriving force arises from the local free-energy difference (Δf) between the FE and AFE\nstates at a given temperature, which represents the functional derivative of the total free\nenergy with respect to the polarization order parameter in the time-dependent Ginzburg-\nLandau (TDGL) equation. Physically, this quantity determines the direction and rate of\ndomain evolution as the system seeks to minimize its total free energy.\nAmong these parameters, the gradient energy coefficient controls the domain-wall\nwidth and energy, whereas the kinetic coefficient (L) governs the dynamic response of\ndomain-wall motion. In the TDGL framework, the diffusion of the domain wall is\njointly determined by the thermodynamic driving force (Δf) and the kinetic coefficient\nL, modulated by the interfacial and elastic energy terms that affect the wall mobility\nand shape. Accordingly, the domain-wall velocity follows a linear relationship with\nthe local free-energy difference and the kinetic coefficient, expressed as [45,\n𝑉\n46], where L represents the temperature-dependent interfacial mobility.\n𝑉 ∝ 𝐿Δ𝐺\nTo capture microstructural effects, localized perturbations were introduced into the\norder parameter field to mimic structural defects such as dislocations or compositional\ninhomogeneities. In the simulations, several grid points were randomly assigned small\nfluctuations (within±5%) in the local polarization, thereby producing spatial variations\nin the free-energy landscape. These perturbations act as pinning centers or nucleation\nsites, leading to curved domain walls, interface bending, and heterogeneous phase\nevolution.\nThe temperature dependence of the domain-wall velocity originates from the explicit\ntemperature dependence of both the Landau coefficient (thermodynamic term) and the\nkinetic coefficient (dynamic term). The Landau coefficient governs the curvature and\ndepth of the double-well potential, thus controlling the free-energy difference between\nthe FE and AFE states, while the kinetic coefficient describes the rate of polarization\nrelaxation. As temperature decreases, the potential wells deepen, increasing the free-\nenergy imbalance (Δf) and accelerating domain-wall motion; however, at lower\ntemperatures, kinetic limitations dominate due to the saturation of L.\nUsing this multiscale model, we performed simulations of the FE-AFE phase transition\nunder low-temperature conditions. Below the critical transition temperature, a local\nstructural perturbation-such as a microcrack, inclusion, or dislocation-was introduced\nto initiate nucleation. As illustrated in Fig. 5a, the antiferroelectric phase preferentially\nnucleates at the perturbation site and subsequently propagates outward, forming a\ncurved domain wall that advances with a diffusion-like velocity V. This process\nprogressively transforms the ferroelectric phase into the antiferroelectric phase over\ntime (Video S2).\nFig. 5e presents the computed domain-wall velocity as a function of temperature,\ndirectly obtained from the nondimensionalized phase-field simulations and then\nconverted to physical units through the kinetic coefficient described above. As shown\nin Fig. S7, the simulations reveal that the lattice-transformation energy barrier gradually\ndecreases as temperature decreases, leading to a larger ΔG and a correspondingly\nstronger thermodynamic driving force. This explains the initial rapid increase of the\nphase-transition velocity at low temperatures (Fig. 5e). However, once the temperature\nfalls below a critical threshold, the energy barrier becomes nearly eliminated, and the\nsystem approaches its minimum-energy configuration. In this regime, the domain-wall\nmotion is no longer limited by the thermodynamic driving force but by intrinsic kinetic\nfactors-such as structural rearrangement and polarization relaxation [45, 46]-which\ncause the kinetic coefficient to saturate. Consequently, the overall domain-wall velocity\nV reaches a limiting value V , representing the intrinsic dynamical limit of the lattice-\nmax\ntransformation process rather than thermodynamic control.\nOverall, these simulations demonstrate that the phase-field model quantitatively links\nfirst-principles energetics with mesoscale domain dynamics, clarifying how the\nmathematical framework governs the physical features observed in Fig. 5, including\nnucleation, defect-induced domain-wall curvature, and temperature-dependent\ninterface velocity. The excellent agreement between the simulated and experimental\ndata demonstrates that the self-consistent coupling of polarization, elastic, and thermal\nfields in the model successfully captures the intrinsic temperature-dependent kinetics\nof the FE-AFE interface migration.\n9. Finally, the English language and grammar of the paper could use some work. I\nwould be happy to suggest changes upon receival of the manuscript modified with the\nquestions above resolved.\nResponse: We sincerely appreciate the reviewer’s kind suggestion. We have carefully\nrevised the manuscript for English language and grammar throughout the text. The\nrevised version has been thoroughly checked to improve clarity and readability. We also\ntruly appreciate the reviewer’s willingness to provide further linguistic advice after\nthese revisions."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have addressed all of my comments and those of the other reviewers in a satisfactory manner and have made\nsubstantial efforts to improve the clarity, quality, and impact of the work. I therefore recommend that the manuscript be\naccepted for publication in its current form.", "reviewer_2": "Dear Authors,\nThank you for the detailed responses to my questions and the changes made to the manuscript. I can now recommend this\narticle for publication, after a few remaining English language changes.\nI went through the first few pages of the manuscript and found some English language problems, which are fixed below:\nLine 26: and the underlying mechanism → and its underlying mechanism\nLine 32-34: Here, we report an irreversible FE-AFE phase transition at low temperatures in Pb(Zr0.97Ti0.03)O3 as seen by\nneutron diffraction patterns\nLine 35: revealed the → reveals that the\nLine 37: revealed the irreversible → reveals that the irreversible\nLine 62: mainly there are two approaches revealed → there are two main approaches revealing\nLine 78: reaveled the → revealed that the\nLine 93: applied to reveal → applied to reveal that\nLine 96: was chosen is due to they → were chosen because they\nThe rest of the manuscript seemed much more grammatically correct, although I didn't review every sentence super closely\nfor grammar.\nBest,"}, "author_response": ""}]

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