*3.4. Quantitative Determination of the Degree of Decomposition During Ball-Milling*

The quantity of lithium oxide segregated at the particle surfaces during ball-milling was determined by coulometric titration in the as-ground samples. No similar measurements were attempted for the annealed samples since the Li2O has a fairly large volatility at the annealing temperature. All titration curves exhibited a single neutralization step as the acid was produced in-situ by the current passing through the cell. This indicates that the primary decomposition product was Li2O and no significant amount of Li2CO3 was present, despite long storing times of several weeks in air elapsed after grinding prior to the titration procedure. The presence of carbonate should have led to a two-stage neutralization process, first leading to bicarbonate formation, but this was never observed. Although the decomposition product detected by the Raman measurement was lithium carbonate, this is no counterargument, as the Raman intensity of the various lithium-containing compounds may be very different and hence, the sensitivity of the Raman measurement may not be comparable for the various possible phases. The weight of dissolved Li2O (*m*OX) was calculated with the following equation:

$$m\_{\rm OX} = MQ / 2F \tag{4}$$

where *M* is the molar weight of Li2O, *Q* is the charge passed until the equivalence point, *F* is the Faraday-constant (96485 C/mol), while the number in the denominator indicates that the hydrolysis of 1 mol of Li2O results in 2 mols of hydroxide ions. Four measurements were performed for each batch ground in different vials. The measured Li2O mass ratios *w*ox, expressed as weight percent of the as-ground powder are given in Table 3. The uncertainty of the measurements is given as the standard deviation of the consecutive titration results.


**Table 3.** Measured and estimated parameters of the nanocrystalline LiNbO3 samples ground in different vials. The weight of the Li2O segregate was measured by titration before annealing, while the LiNb3O8 shell crystallized only upon annealing.

\* Alumina contamination neglected. \*\* Taken for convenience from Table 2. \*\*\* Smaller peak in the size distribution neglected.

The total surface of particles in the sample is proportional to 1/*R*, where *R* is the average particle radius as measured by DLS. The values of *w*OX in Table 3 indeed increase monotonously with 1/*R*, though a fully quantitative trend cannot be established. No similar trend related to the inverse grain radius 1/*r* can be seen in the given range of *r* values obtained by XRD. These observations can be understood if segregation mainly occurs on the outer particle surfaces where both Li2O and O2 may freely leave, giving rise to a niobium-rich layer. However, it is also possible that part of the newly created surfaces, together with part of the Li2O formed, gets buried during later stages of milling and cannot be dissolved for titration.

The quantitative determination of the lithium oxide loss enables us to estimate the thickness of the lithium triniobate layer in the oxidized samples. We use a simplified model of compact, uniform, spherical LN particles of unmodified composition covered by a uniformly thick LiNb3O8 phase. The shell thickness is calculated with the assumption that only the Li2O equivalent to this outer shell could be dissolved and analyzed by titration.

From the reaction indicated in Equation (1) it follows that

$$
\mathfrak{m}\_{\text{OX}} = \mathfrak{m}\_{\text{LTN}} \tag{5}
$$

where *n* stands for the molar quantity of the relevant materials, while the indices OX and LTN refer to the lithium oxide segregate and the lithium triniobate shell of the particles, respectively. For the weight of the particle shell we obtain

$$m\_{\rm OX} = m\_{\rm LTN} \, M\_{\rm OX} / M\_{\rm LTN} \tag{6}$$

where *M* is the molar weight. The weight ratio of the lithium oxide in the ground material, *w*OX, is as follows:

$$w\_{\rm OX} = \frac{m\_{\rm LTN} \frac{M\_{\rm OX}}{M\_{\rm LTN}}}{m\_{\rm LT} + m\_{\rm LTN} \left(1 + \frac{M\_{\rm OX}}{M\_{\rm LTN}}\right)}\tag{7}$$

The weight of each particle component can be expressed with the geometric parameter of the core-shell structure, *d* being the shell thickness and ρ the density:

$$m\_{\rm LN} = \frac{4}{3}\pi (R - d\_{\rm LTN})^3 \rho\_{\rm LN} \approx \frac{4}{3}\pi R^3 \rho\_{\rm LN} - 4\pi R^2 d\_{\rm LTN} \rho\_{\rm LN} \tag{8}$$

$$
\rho\_{\rm LTN} \approx \ 4\pi (R - d\_{\rm LTN})^2 d\rho\_{\rm LTN} \approx 4\pi R^2 d\_{\rm LTN} \rho\_{\rm LTN} \tag{9}
$$

*Crystals* **2019**, *9*, 334

The higher order terms with respect to *d*LTN have been neglected since *d*LTN *R*. The weight ratio of the lithium oxide is then

$$w\_{\rm OX} = \frac{d\_{\rm LTN} \rho\_{\rm LTN} \frac{M\_{\rm COX}}{M\_{\rm LTN}}}{\frac{R}{\mathcal{S}} \rho\_{\rm LN} + d\_{\rm LTN} \left[\rho\_{\rm LTN} \left(1 + \frac{M\_{\rm OX}}{M\_{\rm LTN}}\right) - \rho\_{\rm LN}\right]}}\tag{10}$$

Again, the *d*LTN *R* relation justifies the neglection of the second term in the denominator, leading to

$$d\_{\rm LTN} \approx \frac{R}{3} \frac{\rho\_{\rm LN}}{\rho\_{\rm LTN}} \frac{M\_{\rm LTN}}{M\_{\rm OX}} w\_{\rm OX} \tag{11}$$

By assuming that the Li2O leaving the particle also forms a similar shell in the as-ground sample, for the thickness of this shell we calculate

$$d\_{\rm OX} \approx \frac{\rho\_{\rm LTN}}{\rho\_{\rm OX}} \frac{M\_{\rm OX}}{M\_{\rm LTN}} \, d\_{\rm LTN} \approx \frac{d\_{\rm LTN}}{5.6} \tag{12}$$

where we take <sup>ρ</sup>OX <sup>≡</sup> <sup>ρ</sup>Li2O <sup>≈</sup> 2.01 g/cm3, <sup>ρ</sup>LN <sup>=</sup> 4.65 g/cm<sup>3</sup> and <sup>ρ</sup>LTN <sup>=</sup> 4.975 g/cm3 [39]. The values of *d*LTN and *d*OX are also included in Table 3 and correspond to a layer thickness of at most a few unit cells.

The same amount of segregate (either Li2O or LiNb3O8), if spread evenly on all grain boundaries, would result in a much thinner layer. Neither the corresponding LiNb3O8 layer would be seen as XRD peaks nor would the equivalent amount of Li2O be readily soluble due to its hindered accessibility.

This finding gives further support to our previous assumption that the processes described by Equations (2) and (3) essentially occur on the outer surfaces. Accordingly, particle and grain size reduction proceeds as long as surfaces freshly broken up during ball milling have enough time to pile up a non-ferroelectric surface layer preventing them from stable recoalescing. Below a certain size limit, depending on the detailed properties of the milling system, this becomes impossible as recoalescence becomes too fast. The thickness of the outer segregate layer apparently has a narrow range defined by a similar requirement of sufficient atmospheric contact of the polar surface.

The proposed formation of the core-shell structure would require direct experimental evidence. However, the particle size achieved by the ball-milling process was too large for a direct transmission electron microscopic study of the particles.

Finally, we remark that the given description corresponds to the surface-screening mechanisms in ferroelectric thin films reviewed by Kalinin [40]. In particular, very similar processes seem to occur in prospective lithium-ion batteries using LiNb3O8 as an anode material [41].

#### **4. Conclusions**

Nano-LN was prepared by ball-milling using a Spex 8000 Mixer Mill with different milling parameters. The resulting particle size has a broad distribution in the range of a few hundred nanometers. Five and 20 h of ball-milling resulted in mean grain sizes of about 60 and 40 nm, respectively. Longer ball-millings do not decrease the particle size but only reduce the grain size. α-Al2O3 contamination was found for the sample ground in alumina vial due to the chemically induced abrasion of the vial and the balls during ball-milling. During the milling process the material suffers partial reduction that leads to a balanced formation of bipolarons and polarons yielding gray color together with Li2O segregation on the open surfaces. Upon high-temperature oxidation, the volatile Li2O phase and the polarons get eliminated and the Li deficiency is accommodated by the formation of a more stable LiNb3O8 shell. Darker or brownish color appearing upon high-temperature reduction is caused by the preferential formation of bipolarons. The Li2O loss was observed to increase with the growing total surface of the particles. The average thickness of the non-ferroelectric surface segregate corresponds to a layer of a few unit cells forming the passivating shell of the particles. These findings provide a comprehensive explanation of the physicochemical behavior of the system during grinding and annealing in different atmospheres.

**Author Contributions:** Conceptualization, L.K. (László Kovács), L.P. and G.C.; methodology, L.K. (Laura Kocsor) and J.G.; formal analysis, L.K. (Laura Kocsor); investigation, L.K. (Laura Kocsor); writing—original draft preparation, L.K. (László Kovács), L.P. and G.C.; visualization, L.K. (László Kovács); supervision, L.P.; project administration, Z.K.; funding acquisition, Z.K.

**Funding:** This research was supported by the National Research, Development and Innovation Fund of Hungary within the Quantum Technology National Excellence Program (Project No. 2017-1.2.1-NKP-2017-00001) and the Ministry of Human Capacities of Hungary within the ELTE University Excellence program (1783-3/2018/FEKUTSRAT).

**Acknowledgments:** The authors are grateful to Gábor Piszter and Levente Illés for the optical diffuse reflectance and EDS measurements, respectively.

**Conflicts of Interest:** The authors declare no conflict of interest.
