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Viehland. Microstructures 2023;3:2023016 https://dx.doi.org/10.20517/microstructures.2023.10 Page 3 of 5

Figure 2. (A) Transmission electron microscopy image showing the presence of tweed-like structures in PMN-35PT. Reproduced with
[8]
permission . Copyright 1995 AIP Publishing LLC. (B) Polarized light PLM (top) and piezo-force PFM microscopy (bottom) images.
Reproduced with permission [13] . Copyright 2005 AIP Publishing LLC. (C) Temperature dependence of the general invariance condition of
equation (3.13). Reproduced with permission [16] . Copyright 2003 AIP Publishing LLC.

wall energies that are stress-accommodating (i.e., tweed). Experimentally, this requires the existence of
structurally heterogeneous regions of nanometer size. The spatial and geometric distribution of the
nanoregions is then controlled by special and general invariant conditions that minimize the excess elastic
energy. In the case of PMN-PT type piezoelectric crystals, averaging over an ensemble of tetragonal polar
nanoregions results in an apparent monoclinic MC-type structure by diffraction. Likewise, averaging over
an ensemble of rhombohedral polar nanoregions results in an apparent monoclinic MA-type structure. In
the monoclinic MA and MC phases, the changes in the lattice parameter with temperature and field are
invariant to the geometric conditions of the adaptive phase theory, as shown in Figure 2C. Both the
structurally homogeneous and heterogenous models require the anisotropy of the polarization direction to
be small. In the homogenous case, the polarization vector rotates at the unit cell level; whereas in the
heterogenous case, there is a change in the distribution of the polar nanoregions between equivalent
orientations.

This brings me to the point that this commentary would like to make. It concerns a brief comparison of
another approach to the heterogeneous concept by Li et al. that was put forward some years after the
adaptive phase theory [18,19] . Let us be upfront and direct, the experimental observations that were cited in the
preceding paragraph could equally provide support to either one of these heterogenous models, as the data
reflects a critical role of the local structure on the average structure-property relations. The concept by
Li et al. and the adaptive phase [15-17] are on the same general page [17,18] . It is only a question of details, many of
which whose importance may not yet be realized, simply due to the predominance of investigations has
focused on a conventional homogenous phase. Both approaches recognize the existence of polarization
gradient terms in the Landau-Ginzburg (LG) phenomenology and the need to relax the elastic energy.
Additionally, both approaches recognize the important role of the contribution of polar nanoregions within
the average anisotropy set by the poling directions. A unique aspect of the adaptive phase theory that the LG
theory cannot explain by itself is the observed special invariant conditions of the crystal lattice parameters.

Data reported by Li et al. shows that the dielectric constant in the poled piezoelectric state becomes strongly
frequency dispersive at temperatures far below (ΔT = 250-350 K) the temperature of the dielectric maximum
(T ≈ 400 K), as shown in Figure 1B [18,19] . This result clearly demonstrates that polar nanoregion
max
contributions begin to freeze out on cooling below 100 K in the poled piezoelectric phase. It is only in the
temperature range between 100 K [Figure 1B] and the macrodomain to polar nanoregion transition (see
Figure 1A) that the dielectric constant appears to be nondispersive. It is believed this enhanced polarization