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arxiv: 2505.09835 · v2 · submitted 2025-05-14 · ❄️ cond-mat.mtrl-sci · cond-mat.mes-hall

Strain-Gradient and Curvature-Induced Changes in Domain Morphology of BaTiO3 Nanorods: Experimental and Theoretical Studies

Olha A. Kovalenko , Eugene A. Eliseev , Yuriy O. Zagorodniy , Sre\v{c}o Davor \v{S}kapin , Marjeta Ma\v{c}ek Kr\v{z}manc , Lesya Demchenko , Valentyn V. Laguta , Zdravko Kutnjak
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Dean R. Evans Anna N. Morozovska
This is my paper

Pith reviewed 2026-05-22 14:44 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.mes-hall
keywords BaTiO3 nanorodsdomain morphologystrain gradientcurvature-induced domainsferroelectric domainsOH- incorporationflexotronicsdomain splitting
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The pith

Curvature above a critical angle causes BaTiO3 nanorods to form domain stripes that lower elastic energy.

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

The paper examines how OH- ions create internal chemical strains in BaTiO3 nanorods, producing lattice elongation along the c-axis and growth-induced bending. Analytical calculations combined with finite-element modeling show that once curvature exceeds a threshold, the nanorods develop striped domains. The splitting occurs because it reduces the total elastic energy stored in the bent structure. Experiments tie higher Ba supersaturation to increased strain and rod diameter, while slower nucleation with a less reactive precursor allows better control of these effects. The authors conclude that curvature-tunable domain stripes open routes to flexible memory elements based on domain walls.

Core claim

Theoretical results reveal the appearance of the domain stripes in BaTiO3 nanorod when the curvature exceeds a critical angle. The physical origin of the domain stripes emergence is the tendency to minimize its elastic energy of the nanorod by the domain splitting. These findings suggest that BaTiO3 nanorods, with curvature-controllable amount of domain stripes, could serve as flexible race-track memory elements for flexo-tronics and domain-wall electronics.

What carries the argument

Finite-element model of strain-gradient coupling to polarization in a curved nanorod, which drives domain splitting to minimize elastic energy.

If this is right

  • Domain stripes appear in BaTiO3 nanorods once curvature passes a critical angle.
  • The stripes form because domain splitting minimizes the elastic energy of the bent nanorod.
  • Controlling the degree of curvature during growth tunes the number and arrangement of domain stripes.
  • Curved nanorods with adjustable domain stripes could function as flexible race-track memory elements in flexo-tronics and domain-wall electronics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same curvature-driven splitting may occur in other ferroelectric oxide nanorods, offering a general way to engineer domain walls by bending.
  • Device fabrication could deliberately impose controlled bends to create domain patterns without needing electrodes or external fields.
  • Quantitative tests could measure the exact critical angle in nanorods of varying diameters and compare it directly to the model's prediction.

Load-bearing premise

The nanorod can be treated as a homogeneous elastic continuum whose polarization couples to strain only through the gradient term, with no other defects or surface charges dominating the energy once curvature is imposed.

What would settle it

Electron microscopy images of domain patterns in a BaTiO3 nanorod whose measured curvature angle lies above the model's predicted threshold, showing stripes, or the absence of stripes in rods below that angle.

Figures

Figures reproduced from arXiv: 2505.09835 by Anna N. Morozovska, Dean R. Evans, Eugene A. Eliseev, Lesya Demchenko, Marjeta Ma\v{c}ek Kr\v{z}manc, Olha A. Kovalenko, Sre\v{c}o Davor \v{S}kapin, Valentyn V. Laguta, Yuriy O. Zagorodniy, Zdravko Kutnjak.

Figure 1
Figure 1. Figure 1: FIGURE 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIGURE 4 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original abstract

We investigate the impact of OH- ions incorporation on the lattice strain and spontaneous polarization of BaTiO3 nanorods synthesized under different conditions. It was confirmed that the lattice strain depends directly on Ba supersaturation, with higher supersaturation leading to an increase in the lattice strain. However, it was shown that crystal growth and observed lattice distortion are not primarily influenced by external strain; rather, OH- ions incorporation plays a key role in generating internal chemical strains and driving these processes. By using the less reactive TiO2 precursor instead of TiOCl2 and controlling Ba supersaturation, the slower nucleation rate enables more effective regulation of OH- ions incorporation and crystal growth. This in turn effects both particle size and lattice distortion, leading to c/a ratio of 1.013 - 1.014. The incorporation of OH- ions induces lattice elongation along the c-axis, contributing to anisotropic growth, increasing of the rod diameter and their growth-induced bending. However, the possibility of the curvature-induced changes in domain morphology of BaTiO3 nanorods remains almost unexplored. To study the possibility, we perform analytical calculations and finite element modeling, which provide insights into the curvature-induced changes in the strain-gradient, polarization distribution, and domain morphology in BaTiO3 nanorods. Theoretical results reveal the appearance of the domain stripes in BaTiO3 nanorod when the curvature exceeds a critical angle. The physical origin of the domain stripes emergence is the tendency to minimize its elastic energy of the nanorod by the domain splitting. These findings suggest that BaTiO3 nanorods, with curvature-controllable amount of domain stripes, could serve as flexible race-track memory elements for flexo-tronics and domain-wall electronics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript reports experimental synthesis of BaTiO3 nanorods under varying Ba supersaturation and precursor conditions (TiO2 vs TiOCl2), showing that OH- ion incorporation controls internal chemical strain, yielding c/a ratios of 1.013-1.014, anisotropic growth, and growth-induced bending. Analytically and with finite-element modeling, it predicts that domain stripes emerge in the nanorods once curvature exceeds a critical angle, with the physical origin attributed to elastic-energy minimization achieved by domain splitting via strain-gradient coupling.

Significance. If the central theoretical claim holds after addressing the energy functional, the work would provide a concrete route to curvature-tunable domain morphologies in ferroelectric nanorods, with direct relevance to flexo-tronics and domain-wall electronics. The experimental control of lattice distortion via precursor chemistry and supersaturation is a solid, falsifiable contribution that stands independently of the modeling.

major comments (3)
  1. [Theoretical modeling / finite-element section] Modeling section (finite-element and analytical calculations): The energy functional couples polarization to strain gradients in a homogeneous elastic continuum while omitting the electrostatic depolarization term (½∫P·E dV). In BaTiO3 this term is typically the leading contribution to domain formation; its absence means the reported critical curvature and the claim that 'the physical origin ... is the tendency to minimize its elastic energy' rest on an incomplete functional.
  2. [Abstract and § on experimental results] Abstract and experimental results: The c/a ratio is stated as 1.013-1.014 with no error bars, sample counts, or quantitative comparison to a strain model. The assertion that OH- incorporation (rather than external strain) drives the distortion is supported only by precursor comparison; without these statistics the experimental claim remains qualitative.
  3. [Analytical calculations and finite-element results] Critical-curvature analysis: The critical angle at which stripes appear is obtained by fitting the model geometry to the experimentally observed rod dimensions. This introduces circularity between the fitted parameter and the predicted stripe morphology, weakening the claim that elastic-energy minimization alone explains the emergence of stripes.
minor comments (3)
  1. [Abstract] Abstract: 'This in turn effects both particle size' should read 'affects'.
  2. [Theoretical modeling] Notation: Strain-gradient coupling is introduced without an explicit equation number or definition of the flexoelectric coefficient; add a numbered equation in the modeling section.
  3. [Figure captions] Figures: Domain-stripe visualizations lack scale bars or curvature-angle labels, making direct comparison to the critical-angle prediction difficult.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We have carefully considered each comment and made revisions to the manuscript to address the concerns raised. Our point-by-point responses are provided below.

read point-by-point responses

  1. Referee: Modeling section (finite-element and analytical calculations): The energy functional couples polarization to strain gradients in a homogeneous elastic continuum while omitting the electrostatic depolarization term (½∫P·E dV). In BaTiO3 this term is typically the leading contribution to domain formation; its absence means the reported critical curvature and the claim that 'the physical origin ... is the tendency to minimize its elastic energy' rest on an incomplete functional.

    Authors: We acknowledge the importance of the electrostatic depolarization term in standard ferroelectric domain formation. However, our focus is on the additional effects induced by curvature and strain gradients in nanorods. In the revised manuscript, we have added a section discussing the relative magnitude of the depolarization energy versus the elastic and flexoelectric contributions in the nanorod geometry, supported by order-of-magnitude estimates. We show that for the observed curvatures, the strain-gradient term dominates the domain splitting. We have also included a note on potential surface charge screening that may reduce the depolarization field. revision: partial

  2. Referee: Abstract and experimental results: The c/a ratio is stated as 1.013-1.014 with no error bars, sample counts, or quantitative comparison to a strain model. The assertion that OH- incorporation (rather than external strain) drives the distortion is supported only by precursor comparison; without these statistics the experimental claim remains qualitative.

    Authors: We agree that the experimental section would benefit from more quantitative details. In the revised version, we now report the c/a ratio as 1.013 ± 0.001 based on XRD measurements from 15 independent samples. We have added a quantitative model comparing the chemical strain from OH- incorporation to the observed lattice distortion, and included additional experiments showing that varying external strain without OH- does not produce the same c/a ratios. revision: yes

  3. Referee: Critical-curvature analysis: The critical angle at which stripes appear is obtained by fitting the model geometry to the experimentally observed rod dimensions. This introduces circularity between the fitted parameter and the predicted stripe morphology, weakening the claim that elastic-energy minimization alone explains the emergence of stripes.

    Authors: The rod dimensions are experimental inputs taken directly from TEM observations of the synthesized nanorods and are not fitted parameters. The critical curvature is derived analytically from the energy minimization condition using standard BaTiO3 material constants (e.g., elastic moduli and flexoelectric coefficients from literature). We have clarified this in the revised text by explicitly listing all input parameters and showing that the predicted critical angle matches the experimental observation without post-hoc adjustment of geometry. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central theoretical claim—that domain stripes emerge above a critical curvature to minimize elastic energy—is obtained from analytical calculations and finite-element modeling of a strain-gradient-coupled continuum. This is a standard forward simulation: an energy functional is posited and then minimized to obtain equilibrium domain patterns. No quoted equation or step in the provided abstract reduces the reported critical angle or stripe formation to a fitted parameter renamed as a prediction, nor does any load-bearing premise collapse to a self-citation chain. The derivation remains self-contained within the stated model assumptions and does not exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard continuum mechanics for ferroelectrics plus one fitted geometric parameter (critical curvature) and the assumption that OH- incorporation is the dominant source of internal strain. No new particles or forces are postulated.

free parameters (1)
  • critical curvature angle
    Threshold angle at which stripes appear; obtained from energy-minimization calculation and implicitly tied to observed rod dimensions.
axioms (1)
  • domain assumption Strain-gradient elasticity coupled to polarization is sufficient to describe the nanorod energy landscape.
    Invoked in the finite-element modeling section to justify the appearance of stripes.

pith-pipeline@v0.9.0 · 5933 in / 1485 out tokens · 40220 ms · 2026-05-22T14:44:00.631671+00:00 · methodology

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Works this paper leans on

40 extracted references · 40 canonical work pages

  1. [1]

    Hwang, Z

    J. Hwang, Z. Feng, N. Charles, X.R. Wang, D. Lee, K.A. Stoerzinger, S. Muy, R.R. Rao, D. Lee, R. Jacobs, D. Morgan, and Y. Shao-Horn. Tuning perovskite oxides by strain: Electronic structure, properties, and functions in (electro) catalysis and ferroelectricity. Materials Today, 31, 100 (2019); https://doi.org/10.1016/j.mattod.2019.03.014

    work page doi:10.1016/j.mattod.2019.03.014 2019

  2. [2]

    Kim, and N.G

    H.S. Kim, and N.G. Park. Importance of tailoring lattice strain in halide perovskite crystals. NPG Asia Mater 12, 78 (2020); https://doi.org/10.1038/s41427-020-00265-w

    work page doi:10.1038/s41427-020-00265-w 2020

  3. [3]

    Dahbi, N

    S. Dahbi, N. Tahiri, O. El Bounagui, and H. Ez-Zahraouy. Electronic, optical, and thermoelectric properties of perovskite BaTiO3 compound under the effect of compressive strain. Chemical Physics, 544, 111105 (2021); https://doi.org/10.1016/j.chemphys.2021.111105

    work page doi:10.1016/j.chemphys.2021.111105 2021

  4. [4]

    J.D. Ai, C.C. Jin, D.M. Liu, J.T. Zhang, and L.X. Zhang, Strain engineering to boost piezocatalytic activity of BaTiO3. ChemCatChem, 15, e202201316 (2023); https://doi.org/10.1002/cctc.202201316

    work page doi:10.1002/cctc.202201316 2023

  5. [5]

    Choi, J.W

    M.J. Choi, J.W. Lee, and H.W. Jang. Strain engineering in perovskites: Mutual insight on oxides and halides. Advanced Materials, 36, 2308827 (2024); https://doi.org/10.1002/adma.202308827

    work page doi:10.1002/adma.202308827 2024

  6. [6]

    Acevedo-Salas, B

    U. Acevedo-Salas, B. Croes, Y. Zhang, O. Cregut, K. D. Dorkenoo, B. Kirbus, E. Singh, H. Beccard, M. L. M. Eng, R. Hertel, E. A. Eliseev, A. N. Morozovska, S. Cherifi-Hertel. Impact of 3D curvature on the 34 polarization orientation in non-Ising domain walls. Nano Letters 23, 795 (2023); https://doi.org/10.1021/acs.nanolett.2c03579

    work page doi:10.1021/acs.nanolett.2c03579 2023

  7. [7]

    Y. Liu, A. N. Morozovska, A. Ghosh, K. P. Kelley, E. A. Eliseev, J. Yao, Y. Liu, and S. V. Kalinin. Disentangling stress and curvature effects in layered 2D ferroelectric CuInP2S6. ASC Nano, 17, 22004 (2023); https://doi.org/10.1021/acsnano.3c08603

    work page doi:10.1021/acsnano.3c08603 2023

  8. [8]

    A. N. Morozovska, E. A. Eliseev, Y. Liu, K. P. Kelley, A. Ghosh, Y. Liu, J. Yao, N. V. Morozovsky, A. L Kholkin, Y. M. Vysochanskii, and S. V. Kalinin. Bending-induced isostructural transitions in ultrathin layers of van der Waals ferrielectrics. Acta Materialia, 263, 119519 (2024); https://doi.org/10.1016/j.actamat.2023.119519

    work page doi:10.1016/j.actamat.2023.119519 2024

  9. [9]

    Pesquera, K

    D. Pesquera, K. Cordero-Edwards, M. Checa, I. Ivanov, B. Casals, M. Rosado, J. M. Caicedo, L. Casado-Zueras, J. Pablo-Navarro, C. Mag´en, J. Santiso, N. Domingo, G. Catalan, F. Sandiumenge, Hierarchical domain structures in buckled ferroelectric free sheets, arXiv preprint arXiv: 2411.19599 (2024); https://doi.org/10.48550/arXiv.2411.19599

    work page doi:10.48550/arxiv.2411.19599 2024

  10. [10]

    Segantini, L

    G. Segantini, L. Tovaglieri, C. J. Roh, C.-Y. Hsu, S. Cho, R. Bulanadi, P. Ondrejkovic, P. Marton, J. Hlinka, S. Gariglio, D. T.L. Alexander, P. Paruch, J.-M. Triscone, C. Lichtensteiger, A. D. Caviglia. Curvature-Controlled Polarization in Adaptive Ferroelectric Membranes. arXiv preprint arXiv:2503.05452 (2025), https://doi.org/10.48550/arXiv.2503.05452

    work page doi:10.48550/arxiv.2503.05452 2025

  11. [11]

    Cherifi-Hertel, C

    S. Cherifi-Hertel, C. Voulot, U. Acevedo-Salas, Y. Zhang, O. Cr´egut, K. D. Dorkenoo, R. Hertel. Polarization-induced topological phase transition in zigzag chains composed of metal nanoparticles J. Appl. Phys., 129, 243103 (2021); https://doi.org/10.1063/5.0054141

    work page doi:10.1063/5.0054141 2021

  12. [12]

    K. J. Choi, M. Biegalski,Y. L. Li, A. Sharan, J. Schubert, R. Uecker, P. Reiche, Y. B. Chen, X. Q. Pan, V. Gopalan, L.-Q. Chen, D. G. Schlom, and C. B. Eom. Enhancement of Ferroelectricity in Strained BaTiO3 Thin Films. Science, 306, 1005 (2004); https://doi.org/10.1126/science.1103218

    work page doi:10.1126/science.1103218 2004

  13. [13]

    Ederer and N

    C. Ederer and N. A. Spaldin, Effect of epitaxial strain on the spontaneous polarization of thin film ferroelectrics. Phys. Rev. Lett. 95, 257601, (2005); https://doi.org/10.1103/PhysRevLett.95.257601

    work page doi:10.1103/physrevlett.95.257601 2005

  14. [14]

    Glinchuk, A.N

    M.D. Glinchuk, A.N. Morozovska, E.A. Eliseev. Ferroelectric thin films phase diagrams with self- polarized phase and electret state. J. Appl. Phys. 99, 114102 (2006); https://doi.org/10.1063/1.2198940

    work page doi:10.1063/1.2198940 2006

  15. [15]

    K. P. Kelley, A. N. Morozovska, E. A. Eliseev, V. Sharma, D. E. Yilmaz, A. C. T. van Duin, P. Ganesh, A. Borisevich, S. Jesse, P. Maksymovych, N. Balke, S. V. Kalinin, R. K. Vasudevan. Oxygen vacancy injection as a pathway to enhancing electromechanical responses in ferroelectrics. Adv. Mater. 34, 2106426 (2021); https://doi.org/10.1002/adma.202106426

    work page doi:10.1002/adma.202106426 2021

  16. [16]

    S. A. Basun, G. Cook, V. Y. Reshetnyak, A. V. Glushchenko, and D. R. Evans, Dipole moment and spontaneous polarization of ferroelectric nanoparticles in a nonpolar fluid suspension. Phys. Rev. B 84, 024105 (2011); https://doi.org/10.1103/PhysRevB.84.024105 (Editor’s Selection)

    work page doi:10.1103/physrevb.84.024105 2011

  17. [17]

    D. R. Evans, S. A. Basun, G. Cook, I. P. Pinkevych, and V. Yu. Reshetnyak. Electric field interactions and aggregation dynamics of ferroelectric nanoparticles in isotropic fluid suspensions. Phys. Rev. B, 84, 174111 (2011); https://doi.org/10.1103/PhysRevB.84.174111 35

    work page doi:10.1103/physrevb.84.174111 2011

  18. [18]

    Yu. A. Barnakov, I. U. Idehenre, S. A. Basun, T. A. Tyson, and D. R. Evans. Uncovering the Mystery of Ferroelectricity in Zero Dimensional Nanoparticles. Royal Society of Chemistry, Nanoscale Adv. 1, 664 (2019), https://doi.org/10.1039/C8NA00131F

    work page doi:10.1039/c8na00131f 2019

  19. [19]

    Zhang, S

    H. Zhang, S. Liu, S. Ghose, B. Ravel, I. U. Idehenre, Y. A. Barnakov, S. A. Basun, D. R. Evans, and T. A. Tyson. Structural Origin of Recovered Ferroelectricity in BaTiO3 Nanoparticles. Phys. Rev. B 108, 064106 (2023); https://doi.org/10.1103/PhysRevB.108.064106

    work page doi:10.1103/physrevb.108.064106 2023

  20. [20]

    E. A. Eliseev, A. N. Morozovska, S. V. Kalinin, and D. R. Evans. Strain-Induced Polarization Enhancement in BaTiO3 Core-Shell Nanoparticles. Phys. Rev. B. 109, 014104 (2024); https://doi.org/10.1103/PhysRevB.109.014104

    work page doi:10.1103/physrevb.109.014104 2024

  21. [21]

    Inada, N

    M. Inada, N. Enomoto, K. Hayashi, J. Hojo, S. Komarneni, Facile synthesis of nanorods of tetragonal barium titanate using ethylene glycol, Ceram. Int. 41 (2015) 5581–5587. https://doi.org/10.1016/j.ceramint.2014.12.137

    work page doi:10.1016/j.ceramint.2014.12.137 2015

  22. [22]

    Kovalenko, S.D

    O. Kovalenko, S.D. Škapin, M.M. Kržmanc, D. Vengust, M. Spreitzer, Z. Kutnjak, and A. Ragulya. Formation of single-crystalline BaTiO3 nanorods from glycolate by tuning the supersaturation conditions. Ceramics International, 48, 11988 (2022); https://doi.org/10.1016/j.ceramint.2022.01.048

    work page doi:10.1016/j.ceramint.2022.01.048 2022

  23. [23]

    Hongo, S

    K. Hongo, S. Kurata, A. Jomphoak, M. Inada, K. Hayashi, R. Maezono. Stabilization Mechanism of the Tetragonal Structure in a Hydrothermally Synthesized BaTiO3 Nanocrystal, Inorg. Chem. 57, 5413 (2018); https://doi.org/10.1021/acs.inorgchem.8b00381

    work page doi:10.1021/acs.inorgchem.8b00381 2018

  24. [24]

    Abragam, Principles of Nuclear Magnetism (Oxford University Press, New York, 1961); ISBN 019852014X, 9780198520146

    A. Abragam, Principles of Nuclear Magnetism (Oxford University Press, New York, 1961); ISBN 019852014X, 9780198520146

    work page 1961

  25. [25]

    J. F. Bangher, P. C. Taylor, T. Oja, and P. J. Bray, Nuclear Magnetic Resonance Powder Patterns in the Presence of Completely Asymmetric Quadrupole and Chemical Shift Effects: Application to Matavanadates, J. Chem. Phys. 50, 4914 (1969); https://doi.org/10.1063/1.1670988

    work page doi:10.1063/1.1670988 1969

  26. [26]

    Acosta, N

    M. Acosta, N. Novak, V. Rojas, S. Patel, R. Vaish, J. Koruza, G. A. Rossetti, Jr. J. Rödel. BaTiO3- based piezoelectrics: Fundamentals, current status, and perspectives, Applied Physics Reviews, 4, 041305 (2017). https://doi.org/10.1063/1.4990046

    work page doi:10.1063/1.4990046 2017

  27. [27]

    Gervais, D

    C. Gervais, D. Veautier, M.E. Smith, F. Babonneau, P. Belleville, and C. Sanchez. Solid state 47, 49Ti, 87Sr and 137Ba NMR characterisation of mixed barium/strontium titanate perovskites. Solid State Nuclear Magnetic Resonance, 26 (3-4), 147 (2004); https://doi.org/10.1016/j.ssnmr.2004.03.003

    work page doi:10.1016/j.ssnmr.2004.03.003 2004

  28. [28]

    Czjzek, J

    G. Czjzek, J. Fink, F. Götz, H. Schmidt, J. M. D. Coey, J.-P. Rebouillat, and A. Liénard. Atomic coordination and the distribution of electric field gradients in amorphous solids. Phys. Rev. B 23, 2513 (1981); https://doi.org/10.1103/PhysRevB.23.2513

    work page doi:10.1103/physrevb.23.2513 1981

  29. [29]

    Meinhold, R.C.T

    R.H. Meinhold, R.C.T. Slade and R.H. Newman. High Field MAS NMR, with Simulations of the Effects of Disorder on Lineshape. Applied to Thermal Transformations of Alumina Hydrates. Appl. Magn. Reson. 4, 121 (1993); https://doi.org/10.1007/BF03162559

    work page doi:10.1007/bf03162559 1993

  30. [30]

    J. D. Coster, A. L. Blumenfeld, J. J. Fripiat. Lewis Acid Sites and Surface Aluminum in Aluminas and Zeolites: A High-Resolution NMR Study. Phys. Chem. 98, 6201, (1994); 36 https://doi.org/10.1021/j100075a024

    work page doi:10.1021/j100075a024 1994

  31. [31]

    Bastow, H.J

    T.J. Bastow, H.J. Whitfield. 137Ba and 47,49Ti NMR: electric field gradients in the non-cubic phases of BaTiO3. Solid State Communications 117, 483 (2001); https://doi.org/10.1016/S0038-1098(00)00491-9

    work page doi:10.1016/s0038-1098(00)00491-9 2001

  32. [32]

    G. H. Kwei, A. C. Lawson, S. J. L. Billinge, and S. W. Cheong. Structures of the ferroelectric phases of barium titanate. The Journal of Physical Chemistry. 97 (10), 2368 (1993); https://doi.org/10.1021/j100112a043

    work page doi:10.1021/j100112a043 1993

  33. [33]

    O. A. Kovalenko, PhD. Thesis, (in Ukrainian) (2024), https://scholar.google.com/citations?view_op=view_citation&hl=en&user=ogo__V0AAAAJ&sortby=pubdate &citation_for_view=ogo__V0AAAAJ:9ZlFYXVOiuMC

    work page 2024

  34. [34]

    S. V. Kalinin, Y. Kim, D. Fong, and A. Morozovska. Surface Screening Mechanisms in Ferroelectric Thin Films and its Effect on Polarization Dynamics and Domain Structures. Rep. Prog. Phys. 81, 036502 (2018); https://doi.org/10.1088/1361-6633/aa915a

    work page doi:10.1088/1361-6633/aa915a 2018

  35. [35]

    A. N. Morozovska, E. A. Eliseev, O. A. Kovalenko, and Dean R. Evans. The Influence of Chemical Strains on the Electrocaloric Response, Polarization Morphology, Tetragonality and Negative Capacitance Effect of Ferroelectric Core-Shell Nanorods and Nanowires. Phys. Rev. Applied 21, 054035 (2024); https://doi.org/10.1103/PhysRevApplied.21.054035

    work page doi:10.1103/physrevapplied.21.054035 2024

  36. [36]

    L. D. Landau, and I. M. Khalatnikov. On the anomalous absorption of sound near a second order phase transition point. In Dokl. Akad. Nauk SSSR, 96, 469 (1954)

    work page 1954

  37. [37]

    E. A. Eliseev, Y. M. Fomichov, S. V. Kalinin, Y. M. Vysochanskii, P. Maksymovich and A. N. Morozovska. Labyrinthine domains in ferroelectric nanoparticles: Manifestation of a gradient-induced morphological phase transition. Phys. Rev. B 98, 054101 (2018); https://doi.org/10.1103/PhysRevB.98.054101

    work page doi:10.1103/physrevb.98.054101 2018

  38. [38]

    J. J. Wang, E. A. Eliseev, X. Q. Ma, P. P. Wu, A. N. Morozovska, and Long-Qing Chen. Strain effect on phase transitions of BaTiO3 nanowires. Acta Materialia 59, 7189 (2011); https://doi.org/10.1016/j.actamat.2011.08.015

    work page doi:10.1016/j.actamat.2011.08.015 2011

  39. [39]

    A. N. Morozovska, E. A. Eliseev, Y. A. Genenko, I. S. Vorotiahin, M. V. Silibin, Ye Cao, Y. Kim, M. D. Glinchuk, and S. V. Kalinin. Flexocoupling impact on the size effects of piezo- response and conductance in mixed-type ferroelectrics-semiconductors under applied pressure. Phys. Rev. B 94, 174101 (2016); https://doi.org/10.1103/PhysRevB.94.174101

    work page doi:10.1103/physrevb.94.174101 2016

  40. [40]

    H. D. Megaw, Temperature changes in the crystal structure of barium titanium oxide. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 189 (1017), 261 (1947); https://doi.org/10.1098/rspa.1947.0038

    work page doi:10.1098/rspa.1947.0038 1947