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Fujii et al. Microstructures 2023;3:2023045 https://dx.doi.org/10.20517/microstructures.2023.43 Page 11 of 14

Figure 11. The ac field dependence of the dielectric properties of model MLCCs with different grain sizes measured at various
temperatures.

It is also notable that the values for α' for these formulated BaTiO ceramics are approximately an order of
3
i
magnitude smaller than those of the undoped BaTiO in the ferroelectric phases. There may be several
3
reasons for this. First, the formulated ceramics had a small grain size of ~0.5 µm, which induced additional
grain boundary dilution of the response. Secondly, the formulated BaTiO has a lower volume fraction of
3
the ferroelectric phase associated with the introduction of shell regions. These non-ferroelectric regions will
also act as capacitors in series that suppress the extrinsic contributions to the response. Thirdly, it is also
possible that the core-shell microstructures act to suppress the long coupling of the domain wall motion. It
has been shown that domain walls often have some level of continuity across grain boundaries and that
[36]
the resulting ensembles of domain walls often move as clusters rather than as isolated walls [37,38] . It is not
straightforward to separate the relative importance of each of these factors since the second and third, in
particular, will also be functions of temperature themselves.

Figure 11 shows the ac field dependence of the dielectric constant for formulated BaTiO MLCCs with
3
various grain sizes measured at various temperatures. Above T , the dielectric constant was independent of
c
the ac field amplitude. However, the permittivity data do not superimpose for the different grain sizes,
[39]
presumably because of a difference in the grain boundary dilution of the response . Below T , the dielectric
c
constant depends on the ac field, and the ac field dependence increases with decreasing temperature. The ac
field dependence was quantified using pseudo-Rayleigh analysis, as shown in Figure 12. The ε' followed a
j
core/shell response of the dielectric constant. With decreasing grain size, the ε' decreased. The temperature
j
dependence of the α' was similar to the formulated BaTiO ceramics. The peak at the orthorhombic to
i
3
rhombohedral phase transition decreased with grain size. This is consistent with the grain size dependence
of the undoped BaTiO ceramics. It was observed that the grain size dependence of α' was relatively weak
3
i
above the T of the shell. This result is consistent with the pO dependence of α' for the formulated BaTiO
3
c
i
2
ceramics, suggesting the influence of the shell region on irreversible domain wall contributions.