Pith. sign in

REVIEW 3 major objections 5 minor 40 references

Percolating Multifractal Domains at a Polymorphic Phase Boundary

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read At a polymorphic phase boundary in a lead-free piezoelectric, the polar structure is a percolating multifractal network, and the largest reversible piezoelectric response occurs near the three-dimensional percolation threshold.

desk verdict Convincing qualitative backbone/compliance mechanism for PPB, but the percolation-criticality claim is not yet supported. read the letter →

arxiv 2607.26586 v1 pith:D7CIB3CC submitted 2026-07-29 cond-mat.mtrl-sci physics.comp-ph

Yuan-Jinsheng Liu , Xu Tian , Jiahui Zhai , Shi Liu This is my paper
classification cond-mat.mtrl-sciphysics.comp-ph
keywords percolationmultifractalpolymorphicphaseboundarypiezoelectricityferroelectricsmoleculardynamicspolardomainslead-freepiezoelectrics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper attempts to establish that the polymorphic phase boundary (PPB) in the lead-free piezoelectric (K,Na)NbO3–(Bi,Na)ZrO3 is not simply a coexistence region of nearly degenerate ferroelectric phases, but hosts a percolating multifractal polar domain. Using large-scale molecular dynamics, the authors show that this fractal network is characterized by a global fractal dimension D0 and a multifractal spectrum width Δα, and that the reversible piezoelectric coefficient d33 reaches a volcano-shaped peak when the fractal-domain volume fraction ωf is approximately 0.371, close to the three-dimensional percolation threshold of 0.312. The proposed mechanism is a division of labor: the fractal backbone preserves the initial polar state and provides restoring force, while the surrounding nonfractal regions rotate easily under field and produce large strain. If correct, the work would replace the phase-coexistence explanation with a connectivity explanation and offer fractal connectivity as a design parameter for high-performance lead-free piezoelectrics.

What carries the argument

The load-bearing object is the polar fractal domain, identified by a domain-growing algorithm that starts from randomly seeded cells with sizable local polarization and adds neighboring cells whose polarization stays within 30° of the domain direction over a 200-ps correlation window. Its structural fingerprints are D0 (global box-counting fractal dimension), Δα (width of the multifractal singularity spectrum, measuring spatial heterogeneity), and ωf (the volume fraction occupied by the fractal domain). The comparison of ωf with the 3D percolation threshold, and the dual-role picture of the fractal backbone (rigid, memory-preserving) versus nonfractal surroundings (compliant, strain-producin

What would settle it

If an independent measure of fractal connectivity (e.g., from X-ray nanodiffraction or PFM imaging across the PPB in KNN-BNZ) showed no percolating fractal polar network with a volume fraction near 0.31–0.37 at the temperature of maximum d33, the central claim would be contradicted. Alternatively, a sensitivity analysis of the domain-growing thresholds that shifts ωf far from 0.312 while retaining the volcano peak would undermine the percolation interpretation.

Watch

Extended reading notes

Core claim

The central discovery is that at T*=140 K, inside the simulated PPB of KNN-BNZ, the polar microstructure forms a single connected, system-spanning network with fractal dimension D0=2.925 and multifractal spectrum width Δα=1.196—a dense but noncompact backbone with tenuous branches. The authors find that this network acts as a relatively rigid polar memory: dipoles inside the fractal domain rotate less than those outside, and the reversible strain–field loop closes because the backbone restores the initial orientation after field removal. Across more than twenty zero-field polar states and a range of temperatures, the effective reversible d33 follows a single 'volcano' when plotted against th

Load-bearing premise

The comparison with the percolation threshold rests on the particular operational definition of what counts as a 'fractal domain' (a growth rule requiring polarization alignment within 30° over 200 ps), and a different cutoff would shift ωf; in addition, the simulated PPB temperature (≈140 K) is mapped to the experimental one by a rescaling that the force field requires.

Editorial extensions

If this is right

  • If the percolating multifractal domain is the microscopic mechanism, then PPB-enhanced giant reversible piezoelectricity is a connectivity effect: near-threshold fractal connectivity enables both reversibility (backbone) and large strain (compliant regions), not simply a flat energy landscape.
  • The global fractal dimension D0 and multifractal width Δα act as microstructural order parameters that can rationalize the temperature-dependent dielectric plateau and the decoupling between dielectric and piezoelectric peaks.
  • The optimal ωf≈0.371 provides a quantitative design target: to maximize reversible d33, materials should be tuned so that the fractal polar network occupies close to the percolation-critical volume fraction.
  • The same composition can show either closed reversible loops or open, trapped loops depending on the initial poling state; this explains the experimentally known poling sensitivity of KNN-based ceramics.
  • The mechanism is generic: fractal connectivity as a design parameter could apply to other structurally disordered systems where reversible large deformation is sought.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Since the paper's ωf is defined via a specific domain-growing protocol with hand-chosen thresholds (30°, 200 ps), a natural next step would be to test how sensitive the volcano peak position and height are to these thresholds; if the peak robustly sits near percolation across reasonable variations, the claim is strengthened.
  • Experimental verification could come from high-resolution reciprocal-space mapping or piezoresponse force microscopy across the PPB: if a percolating fractal polar network with volume fraction ≈0.35 is not observed at the temperature of maximum d33, the connectivity explanation would be in doubt.
  • The temperature rescaling required by the force field implies the simulated T*=140 K corresponds to a higher experimental temperature; if the fractal connectivity criterion is universal, the same volcano relationship may hold when d33 is measured as a function of temperature in a single sample, making ωf a hidden variable that collapses data onto one curve.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper uses large-scale molecular dynamics with a machine-learned force field (UniPero) to study the polymorphic phase boundary (PPB) in lead-free KNN–BNZ. It identifies a connected, system-spanning polar domain at T* = 140 K, characterized by a global fractal dimension D0 = 2.925 and a multifractal spectrum width Δα = 1.196. The authors propose that this 'fractal domain' acts as a rigid backbone preserving polar memory, while surrounding nonfractal regions provide compliance, enabling large reversible piezoelectric response. They report d33 = 809.8 pm/V for the [111] initial state, with closed-loop reversibility, whereas the [221] state gives an open loop and d33 = 418.7 pm/V. By pooling states and temperatures, they construct a volcano-shaped plot of d33 versus fractal-domain volume fraction ωf, peaking at ωf ≈ 0.371, which they claim is close to the 3D percolation threshold of 0.312. The central claim is that near-critical percolating connectivity of a multifractal polar domain is the microscopic mechanism for PPB-enhanced piezoelectricity.

Significance. If the percolation-criticality claim were quantitatively established, this would be a conceptually novel mechanism for PPB-enhanced piezoelectricity, moving beyond the standard phase-coexistence picture and offering a design parameter (fractal connectivity) for lead-free piezoelectrics. The qualitative backbone/compliance mechanism is well supported by several internal controls: the closed-loop versus open-loop contrast between [111] and [221] states, the consistent inside/outside rotation-angle asymmetry across more than 20 polar states (Fig. 3f), and the temperature evolution of D0 and Δα. The paper also provides reproducible computational methodology (LAMMPS, a machine-learned potential, explicit finite-size supercell) and validates against experimental dielectric spectra and strain loops. However, the load-bearing quantitative claim—that maximum d33 occurs near the percolation threshold—is not supported by the presented analysis. The volume fraction ωf is not a percolation order parameter, no finite-size scaling or spanning-probability analysis is reported, and the reported D0 = 2.925 is inconsistent with a 3D critical percolation cluster (D ≈ 2.53). These issues are fixable bu

major comments (3)
  1. [Abstract; Fig. 4; Supplementary Sect. II.C] The central claim that maximum reversible d33 occurs near the 3D percolation threshold is not supported by the data. ωf is the volume fraction of the single connected component identified by the custom domain-growing protocol (seed threshold, 30° angular tolerance, 200-ps correlation window), not an occupation probability in a percolation model. The protocol by construction outputs a connected, system-spanning object, so its volume fraction is a cluster property, not a percolation probability. The comparison 0.371 versus 0.312 therefore has no quantitative meaning without a spanning-probability curve, cluster-size distribution, or finite-size scaling analysis, none of which is reported. Moreover, D0 = 2.925 at T* = 140 K (Fig. 2b) is close to the embedding dimension 3, whereas the incipient infinite cluster of 3D percolation has fractal dimension ≈2.53. This indicates a compact, supercri
  2. [Supplementary Sect. II.C; Figs. 2–4] All structural descriptors—D0, Δα, ωf, and the inside/outside partition used in Fig. 3(e)–(f)—depend on hand-chosen thresholds in the domain-growing protocol: the local-polarization seed threshold, the 30° angular tolerance, and the 200-ps correlation window. No sensitivity analysis is reported. Since the volcano peak position in Fig. 4 and the 'close to 0.312' comparison are derived from a single operational definition, the quantitative conclusions are conditional on this protocol. The authors should demonstrate that the main findings are robust over a range of thresholds, or provide a principled, data-driven criterion for selecting them. Without this, the percolation-threshold comparison and the exact value ωf ≈ 0.371 are not meaningful.
  3. [Fig. 4; Fig. 3(b)–(c)] The volcano plot pools data from different temperatures and polar states, but only reversible polarization-rotation paths are included, with the field direction chosen so that the path avoids trapping by other global states. Because reversibility is established using the same simulations that define the fractal domain, and because the field-direction selection is not specified quantitatively, the dataset may be biased toward favorable paths. Error bars, the number of states per temperature, and the criterion for choosing the field direction are not given. This information is necessary to assess whether the peak at ωf ≈ 0.371 is robust or an artifact of the reversible-path selection.
minor comments (5)
  1. [Fig. 1(c); page 4] The effective d33 is said to be extracted 'from the slope' of the strain–electric-field loop, but a loop has field-dependent slope; please specify the field range or the convention (e.g., unipolar slope near zero field or at Emax) used for the reported values.
  2. [Page 4; Discussion of temperature rescaling] The manuscript correctly notes that MD underestimates transition temperatures and invokes a temperature rescaling to compare with experiments. This is an acknowledged limitation, but the mapping is not quantified. Please state the rescaling factor or show that the qualitative conclusions are insensitive to the shift.
  3. [Fig. 2(b); caption] D0 = 2.925 is reported to four significant figures without an uncertainty estimate from the box-counting fit. A standard error or confidence interval would help the reader judge whether D0 is statistically distinct from 3.
  4. [Page 6; Fig. 3(f)] The text mentions 'more than 50 distinct directions' while Fig. 3(f) reports 'more than 20' states. Reconcile these numbers and report the exact sample size in the figure or text.
  5. [References] Minor typos: Ref. [7] 'Naure' should be 'Nature'; Ref. [30] 'molecular synamics' should be 'molecular dynamics'. Also, 'KNN–NBZ' appears once in the text on page 3 and should be 'KNN–BNZ' for consistency.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: d33, D0, Δα, and ωf are independently computed observables, and the percolation-threshold comparison is an external benchmark rather than a fitted input.

full rationale

The paper's derivation chain is: (1) UniPero MD reproduces the experimental PPB dielectric spectrum and giant d33; (2) a connected polar domain, grown by the 30-degree/200-ps correlation rule, is characterized as fractal/multifractal; (3) field-induced rotation angles are smaller inside than outside this domain; (4) reversible d33 plotted against ωf shows a volcano peak near ωf ≈ 0.371, which is compared with the external 3D percolation threshold 0.312 from Ref. [20]. None of these steps defines the piezoelectric response in terms of the domain descriptor or vice versa. ωf, D0, and Δα are computed from zero-field polar structure, while d33 comes from independent strain-electric-field loops; the volcano trend is an observed correlation, not a parameter fitted to d33 and then renamed a prediction. The only self-citations (UniPero [21], Curie-temperature benchmark [29]) are backed by external experimental and ab initio benchmarks, so they are not a closed self-citation loop. The authors explicitly note that MD underestimates phase-transition temperatures and invoke a temperature rescaling; this is an acknowledged limitation affecting quantitative T* mapping, not a circular step. Criticisms that ωf is not an occupation probability in percolation theory, that no finite-size scaling is given, and that D0 = 2.925 is inconsistent with a critical percolation cluster are scientific validity concerns, not demonstrations that a result is equivalent to its input by construction. Therefore no significant circularity is identified.

Assumptions & free parameters 6 free parameters · 4 assumptions · 1 invented entities

The central claim rests on six hand-chosen or fitted parameters, four domain assumptions, and one conceptual entity. The heaviest burden is the domain-growing protocol (30°/200-ps thresholds): it defines the very object whose volume fraction is then compared with the percolation threshold. The temperature rescaling is an acknowledged shift of about 3× that licenses mapping the simulated 140 K state onto the experimental PPB. No free parameter is hidden in the multifractal formalism itself, but the box-counting range on a 32³ supercell is short. In aggregate, the paper contributes a mechanism hypothesis plus an empirical volcano correlation; the percolation-criticality linkage is an interpretation attached after the fact.

free parameters (6)
  • domain-growing angular threshold = 30°
    Hand-chosen cutoff for incorporating neighboring cells into a polar domain; controls which cells count as 'fractal domain,' hence D0, Δα, and ωf. No sensitivity analysis provided.
  • domain-growing correlation time = 200 ps
    Hand-chosen temporal window over which polarization directions must remain within 30°; similarly controls domain identity and the resulting descriptors.
  • local polarization seed threshold = not specified in main text
    'Sizable local polarization' is used as the seed/growth criterion, but its numerical value is not given; the constructed domain depends on it.
  • maximum applied field Emax = 60 kV/cm
    Fixed field amplitude chosen by the authors; the 'reversible' classification of each polar state depends on this value.
  • temperature rescaling = sim T* ≈ 140 K ↔ experimental PPB (~420 K)
    The MD PPB is found at 140 K while the experimental PPB is at much higher temperature; the paper invokes a global 'temperature rescaling' to map them, an implicit shift parameter.
  • optimal volume fraction ωf* = 0.371
    Location of the d33-vs-ωf volcano peak read off Fig. 4; this fitted maximum is then compared with the percolation threshold 0.312.
assumptions (4)
  • domain assumption UniPero MLFF accurately describes KNN-BNZ polar physics at simulated temperatures, after a global temperature rescaling.
    The entire PPB phenomenology is produced by this force field (Ref. 21, same group), validated against experimental dielectric spectra and ab initio energies, but with an acknowledged approximately 3× temperature shift (p. 4).
  • domain assumption The domain-growing protocol (30°, 200-ps criterion) identifies the physically relevant polar domains.
    D0, Δα, and ωf are computed on domains defined by this protocol; the percolation comparison inherits this definition. Thresholds appear in Supplementary Sect. II.C.
  • domain assumption Box-counting and multifractal formalism on a 32×32×32 supercell yields reliable D0 and Δα.
    The scaling range is limited (box sizes over roughly three decades on a 32³ cell); standard formalism from Refs. 31-35 applied to a single MD trajectory.
  • ad hoc to paper The 3D percolation threshold 0.312 for random isotropic systems applies to the polar domain network.
    No percolation analysis (spanning probability, cluster size statistics) is performed on the actual lattice; the literature constant from Ref. 20 is invoked for comparison with a domain defined by the paper's own thresholds.
invented entities (1)
  • percolating multifractal polar domain as a microstructural order parameter (D0, Δα, ωf)
    purpose: Explains PPB-enhanced dielectric plateau and giant reversible piezoelectricity as a near-percolation-critical polar network with a rigid backbone and compliant surroundings.
    The domain is defined by the paper's own growing rules; its descriptors are computed from the same simulations used to compute d33. The only external handle, the percolation threshold 0.312, is invoked post-hoc and without error bars.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Percolating Multifractal Domains at a Polymorphic Phase Boundary." pith.science (2026). https://pith.science/paper/D7CIB3CC

@misc{pith2026260726586,
  author       = {Pith},
  title        = {Pith review of: Percolating Multifractal Domains at a Polymorphic Phase Boundary},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D7CIB3CC}},
  note         = {Machine review of arXiv:2607.26586}
}
read the original abstract

Giant piezoelectricity in ferroelectrics is commonly associated with phase-boundary instabilities, among which the polymorphic phase boundary (PPB) is a prominent example conventionally attributed to the coexistence of ferroelectric phases. Here, using large-scale molecular dynamics simulations of the lead-free (K,Na)NbO3-(Bi,Na)ZrO3 solid solutions, we show that the PPB hosts a percolating multifractal polar domain which governs the dielectric and piezoelectric responses. By quantifying the global fractal dimension and multifractal spectrum width of this polar network, we identify fractal connectivity and multiscale heterogeneity as microstructural order parameters for the PPB. The maximum reversible piezoelectric response occurs when the fractal-domain volume fraction approaches the three-dimensional percolation threshold, suggesting that near-critical polar connectivity enables giant reversible electromechanical coupling. In this mechanism, the fractal backbone preserves polar memory and provides the restoring force required for reversibility, while the surrounding nonfractal regions supply the polar compliance needed for large polarization rotation and strain. These results establish percolating multifractal polar domains as a microscopic mechanism for PPB-enhanced piezoelectricity and suggest fractal connectivity as a design parameter for high-performance piezoelectrics.

Figures

Figures reproduced from arXiv: 2607.26586 by the authors.

Figure 1
Figure 1. MD simulations of the polymorphic phase boundary in lead-free KNN-BNZ solid solutions. [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Polar multifractal domains at the PPB. (a) Polar multifractal domain identified at [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Fractal-domain-assisted reversible giant piezoelectricity at the PPB. (a) Representative [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Volcano plot of d33 as a function of the fractal-domain volume fraction, ωf . The piezoelectric coefficients are computed for various polarization states over a range of temperatures. The volcano￾shaped trend peaks at ωf ≈ 0.371, where d33 exceeds 800 pm/V. This optima…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

40 extracted references · 1 canonical work pages

  1. [1]

    F. Li, B. Wang, X. Gao, D. Damjanovic, L.-Q. Chen, and S. Zhang, Ferroelectric materials toward next-generation electromechanical technologies, Science389, eadn4926 (2025)

    2025

  2. [2]

    R¨ odel, W

    J. R¨ odel, W. Jo, K. T. P. Seifert, E.-M. Anton, T. Granzow, and D. Damjanovic, Perspective on the development of lead-free piezoceramics, J. Am. Ceram. Soc.92, 1153 (2009)

    2009

  3. [3]

    Fu and R

    H. Fu and R. E. Cohen, Polarization rotation mechanism for ultrahigh electromechanical response in single-crystal piezoelectrics, Nature403, 281 (2000)

    2000

  4. [4]

    J. Wu, D. Xiao, and J. Zhu, Potassium–sodium niobate lead-free piezoelectric materials: Past, present, and future of phase boundaries, Chem. Rev.115, 2559 (2015)

    2015

  5. [5]

    Jaffe, W

    B. Jaffe, W. R. Cook, and H. Jaffe,Piezoelectric Ceramics(Academic Press, 1971)

    1971

  6. [6]

    Noheda, D

    B. Noheda, D. E. Cox, G. Shirane, J. A. Gonzalo, L. E. Cross, and S.-E. Park, A monoclinic ferroelectric phase in the Pb(Zr 1−xTix)O3 solid solution, Appl. Phys. Lett.74, 2059 (1999)

  7. [7]

    Ahart, M

    M. Ahart, M. Somayazulu, R. E. Cohen, P. Ganesh, P. Dera, H. K. Mao, R. J. Hemley, P. Ren, Y. a nd Liermann, and Z. G. Wu, Origin of morphotropic phase boundaries in ferroelectrics, Naure 451, 545 (2008)

    2008

  8. [8]

    Park and T

    S.-E. Park and T. R. Shrout, Ultrahigh strain and piezoelectric behavior in relaxor based ferro- electric single crystals, J. Appl. Phys.82, 1804 (1997)

    1997

Show all 40 references
  1. [9]

    Damjanovic, Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics, Rep

    D. Damjanovic, Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics, Rep. Prog. Phys.61, 1267 (1998)

    1998

  2. [10]

    F. Li, S. Zhang, Z. Xu, X. Wei, J. Luo, and T. R. Shrout, Temperature independent shear piezo- electric response in relaxor-PbTiO 3 based crystals, Appl. Phys. Lett.97, 252903 (2010)

    2010

  3. [11]

    Zhang, R

    S. Zhang, R. Xia, and T. R. Shrout, Lead-free piezoelectric ceramics vs. PZT?, J. Electroceram. 19, 251 (2007)

    2007

  4. [12]

    J. Li, K. Wang, F. Zhu, L. Cheng, and F. Yao, (K,Na)NbO 3 -based lead-free piezoceramics: Fun- damental aspects, processing technologies, and remaining challenges, J. Am. Ceram. Soc.96, 3677 (2013)

    2013

  5. [13]

    X. Lv, N. Zhang, J. Wu, and X.-x. Zhang, The role of adding Bi 0.5A0.5ZrO3 in affecting orthorhombic-tetragonal phase transition temperature and electrical properties in potassium sodium niobate ceramics, Acta Mater.197, 224 (2020)

    2020

  6. [14]

    X. Lv, X. Wang, X. Shi, J. Kong, H.-b. Huang, T. Hu, Z. Fu, J. Lyu, Y. Ma, X.-x. Zhang, B. Wu, 10 A. Pramanick, and J. Wu, Beyond phase boundaries: atomic mechanisms governing structure and property variations in (K, Na)NbO 3-based ferroelectrics, Nat. Commun.17, 859 (2025)

    2025

  7. [15]

    Liu and X

    W. Liu and X. Ren, Large piezoelectric effect in Pb-free ceramics, Phys. Rev. Lett.103, 257602 (2009)

    2009

  8. [16]

    D. A. Ochoa, G. Esteves, J. L. Jones, F. Rubio-Marcos, J. F. Fern´ andez, and J. E. Garc ´ ıa, Extrinsic response enhancement at the polymorphic phase boundary in piezoelectric materials, Appl. Phys. Lett.108, 142901 (2016)

    2016

  9. [17]

    D. Wang, M. Cao, and S. Zhang, Phase diagram and properties of Pb(In 1/2Nb1/2)O3– Pb(Mg1/3Nb2/3)O3–PbTiO3 polycrystalline ceramics, J. Eur. Ceram. Soc.32, 433 (2012)

    2012

  10. [18]

    J. E. Garcia and F. Rubio-Marcos, Polymorphic phase boundary in piezoelectric oxides, J. Appl. Phys.127, 131102 (2020)

    2020

  11. [19]

    D. Wang, F. Hussain, A. Khesro, A. Feteira, Y. Tian, Q. Zhao, and I. M. Reaney, Composi- tion and temperature dependence of structure and piezoelectricity in (1−x)(K 1−yNay)NbO3 − x(Bi1/2Na1/2)ZrO3 lead-free ceramics, J. Am. Ceram. Soc.100, 627 (2016)

    2016

  12. [20]

    C. D. Lorenz and R. M. Ziff, Universality of the excess number of clusters and the crossing proba- bility function in three-dimensional percolation, J. Phys. A: Math. Gen.31, 8147 (1998)

    1998

  13. [21]

    J. Wu, J. Yang, Y.-J. Liu, D. Zhang, Y. Yang, Y. Zhang, L. Zhang, and S. Liu, Universal interatomic potential for perovskite oxides, Phys. Rev. B108, L180104 (2023)

    2023

  14. [22]

    Y. Hu, J. Yang, and S. Liu, Giant piezoelectric effects of topological structures in stretched ferro- electric membranes, Phys. Rev. Lett.133, 046802 (2024)

    2024

  15. [23]

    Y.-J. Liu, T. C. Sterling, and S. Liu, Dipolar nematic state in relaxor ferroelectrics, Phys. Rev. X 16, 021022 (2026)

    2026

  16. [24]

    J. Wu, J. Yang, L. Ma, L. Zhang, and S. Liu, Modular development of deep potential for complex solid solutions, Phys. Rev. B107, 144102 (2023)

    2023

  17. [25]

    Y. Shi, R. He, B. Zhang, and Z. Zhong, Revisiting the phase diagram and piezoelectricity of lead zirconate titanate from first principles, Phys. Rev. B109, 174104 (2024)

    2024

  18. [26]

    S. Liu, I. Grinberg, H. Takenaka, and A. M. Rappe, Reinterpretation of the bond-valence model with bond-order formalism: An improved bond-valence-based interatomic potential for PbTiO 3, Phys. Rev. B88, 104102 (2013)

    2013

  19. [27]

    Gindele, A

    O. Gindele, A. Kimmel, M. G. Cain, and D. Duffy, Shell model force field for lead zirconate titanate Pb(Zr1−xTix)O3, J. Phys. Chem. C119, 17784 (2015). 11

    2015

  20. [28]

    J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Restoring the density-gradient expansion for exchange in solids and surfaces, Phys. Rev. Lett.100, 136406 (2008)

    2008

  21. [29]

    D. Li, C. S. Ahart, and S. Liu, Disentangling the discrepancy between theoretical and experimental curie temperatures in ferroelectric PbTiO 3, MRS Commun. 10.1557/s43579-026-01002-0 (2026)

    2026 doi

  22. [30]

    Plimpton, Fast parallel algorithms for short-range molecular synamics, J

    S. Plimpton, Fast parallel algorithms for short-range molecular synamics, J. Comput. Phys.117, 1 (1995)

    1995

  23. [31]

    Mandelbrot, How long is the coast of britain? statistical self-similarity and fractional dimension, Science156, 636 (1967)

    B. Mandelbrot, How long is the coast of britain? statistical self-similarity and fractional dimension, Science156, 636 (1967)

    1967

  24. [32]

    Feder,Fractals(Springer US, 1988)

    J. Feder,Fractals(Springer US, 1988)

    1988

  25. [33]

    Catalan, H

    G. Catalan, H. B´ ea, S. Fusil, M. Bibes, P. Paruch, A. Barth´ el´ emy, and J. F. Scott, Fractal dimension and size scaling of domains in thin films of multiferroic BiFeO3, Phys. Rev. Lett.100, 027602 (2008)

    2008

  26. [34]

    T. C. Halsey and M. H. Jensen, Spectra of scaling indices for fractal measures: Theory and exper- iment, Phys. D: Nonlinear Phenom.23, 112 (1986)

    1986

  27. [35]

    T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A33, 1141 (1986)

    1986

  28. [36]

    Zheng and J

    T. Zheng and J. Wu, Relationship between poling characteristics and phase boundaries of potas- sium–sodium niobate ceramics, ACS Appl. Mater. Interfaces8, 9242 (2016)

    2016

  29. [37]

    Y. Liu, B. Shen, J. Fan, X. Qi, E. Sun, and R. Zhang, Dynamic hysteresis relation for guiding poling condition of high-performance KNN-based ceramics, J. Eur. Ceram. Soc.43, 4044 (2023)

    2023

  30. [38]

    B. Wu, H. Tao, J. Ma, L. Zhao, D. Ergu, L. Luo, W. Wu, and Y. Yao, Facilitated reversible domain switching at multiphase boundary in periodically orthogonal poled KNN-based ceramics: Strain versus non-180◦ domain, J. Eur. Ceram. Soc.107, 1009 (2023)

    2023

  31. [39]

    Y. Huan, X. Wang, J. Koruza, K. Wang, K. G. Webber, Y. Hao, and L. Li, Inverted electro-mechanical behaviour induced by the irreversible domain configuration transformation in (K, Na)NbO3-based ceramics, Sci. Rep.6, 22053 (2016)

    2016

  32. [40]

    B. Wu, L. Zhao, J. Feng, Y. Zhang, X. Song, J. Ma, H. Tao, Z. Xu, Y.-X. Liu, S. Wang, J. Lu, F. Zhu, B. Han, and K. Wang, Contribution of irreversible non-180°domain to performance for multiphase coexisted potassium sodium niobate ceramics, Nat. Commun.15, 2408 (2024). 12 Figu...

    2024

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.