Structural Disorder of Mechanically Activated δ-MgCl₂ Studied by Synchrotron X-ray Total Scattering and Vibrational Spectroscopy

Abstract

A combination of synchrotron X-ray total scattering and molecular simulation is a powerful approach for reliable determination of the structure of δ-MgCl₂ as an indispensable component of heterogeneous Ziegler–Natta catalysts. Here, the same approach is applied to mechanically activated MgCl₂. Four types of mechanically activated MgCl₂ samples are prepared using ball-milling in the absence and presence of different donors. The development of structural disorder along the grinding time is compared. It was found that the presence of donors accelerates the formation of δ-MgCl₂ in an early stage of grinding, while elongated grinding eventually results in δ-MgCl₂ with similar extents of structural disorder in the absence and presence of different donors. The FT-IR investigation consistently verified the morphological similarity between the firmly ground samples. Thus, the structure of δ-MgCl₂ is likely governed by mechanical energy when sufficiently ground.

1. Introduction

MgCl₂-supported Ziegler–Natta catalysts (ZNCs) have played a central role in the production of stereoregular polypropylene for decades [1]. For preparing a performant ZNC, the MgCl₂ support has to be “activated”. The activated support is generally termed as δ-MgCl₂ and featured with a broad powder X-ray diffraction (PXRD) pattern [2,3,4]. It has been considered that δ-MgCl₂ corresponds to nanosized crystallites (below ~30 nm) that are featured with rotational faults in the stacking of Cl–Mg–Cl layers [5,6,7,8]. However, such a consideration has solely relied on an interpretation of featureless PXRD patterns, and it has not been cross-validated with other analytical methods. For example, the solid-state NMR approach [9,10] has successfully distinguished bulk and nanosized MgCl₂, but the variation of the samples was limited, and the ability toward complementary analysis against PXRD is still to be improved. Recently, many computational studies treated MgCl₂ nanoparticle models instead of the infinite surfaces [11,12,13]; however, they are generally far smaller than the experimentally determined dimensions, and the structural characters other than the surfaces were hardly discussed. Therefore, the possibility of other structural disorders that can cause a similarly diffuse PXRD pattern, such as the local atomic displacement from the ideal coordination places, lattice distortion, or another concept of amorphous-like structures [14], could not be excluded.

In recent publications [4,15], we revisited the identity of δ-MgCl₂ with the aid of cutting-edge analytical methods in terms of crystalline structure as well as surface morphology. In particular, the synchrotron X-ray total scattering technique was first applied to the structural investigation of δ-MgCl₂. The X-ray total scattering measurement enables us to access the ordered structure information (PXRD pattern) and the real space information via atomic pair distribution function (PDF). PDF directly shows local structure of atoms below sub-nanometers to medium or long distances of approximately 1 to 100 nm [16,17,18,19], which is exactly relevant for δ-MgCl₂. The PXRD pattern and the PDF were acquired through the X-ray total scattering measurement for a fourth-generation Ziegler–Natta catalyst. They were independently used to derive molecular models. Comparison between the two best-fit models revealed the structure of δ-MgCl₂ in a quantitative and cross-validated manner. It was concluded that the structure of δ-MgCl₂ is described by disorderly stacked MgCl₂ nanoplates. Moreover, vibrational spectroscopies and DFT calculations were applied on the same set of samples, and the obtained morphological insight into the activated δ-MgCl₂ was well consistent with the structural analysis results.

Here, we have adopted the same strategy to reveal the formation of δ-MgCl₂ and the involvement of donors therein: The structural evolution of MgCl₂ was quantitatively investigated during mechanical activation in the presence or absence of different donors by means of the synchrotron X-ray total scattering as well as vibrational spectroscopies. The structure of δ-MgCl₂ in the obtained samples was quantitatively determined with the aid of simulation using MgCl₂ nanoparticle models. The effect of donors was discussed on the formation of δ-MgCl₂. The mechanical activation is rather outdated and rarely applied in recent industry. However, it is suitable as the first step to clarify the formation of δ-MgCl₂ due to its chemical simplicity as well as to the ease of controlling the activation process through the grinding time. As the modern ZNCs possess a high performance as a consequence of the complex interaction among various components over atomic to macroparticle scale [20], the accumulation of fundamental knowledge is important for not only academia, but also industry. In such a sense, the effect of donors on the formation of δ-MgCl₂ is one of the issues that must be clarified. To the best of our knowledge, nobody has quantitatively determined the structure of mechanically activated δ-MgCl₂.

2. Results and Discussion

Mechanically activated δ-MgCl₂ samples were prepared by ball-milling with different grinding durations (3, 6, 12, 24, and 48 h) in a dry condition. To investigate the effect of donors on mechanical activation, the other two series of samples were prepared in the presence of typical donors ethylbenzoate (EB) and di-n-buthylphthalate (DBP). The amount of donors was set to the molar ratio of MgCl₂:donor = 16:1 mol/mol [20]. It is considered that donors play two distinct roles in mechanical activation [21]: (i) Capping effect by adsorbing on coordinatively unsaturated Mg²⁺ ions that are newly exposed in the course of grinding, and (ii) physical effects such as absorption of collision energy as a liquid and adhering particles by wetting their surfaces. Taking this speculation into account, using an inert liquid additive instead of donors would be worthwhile in terms of distinguishing the two roles. Toluene was chosen as the inert additive as it has no functional groups that can adsorb on Mg²⁺ ions while it contains an aromatic ring similar to EB and DBP. We thus prepared four series of samples (Dry-xx, EB-xx, DBP-xx, and Tol-xx; xx is the grinding duration in hours) to be subjected to synchrotron X-ray total scattering measurement. The derived high-quality PXRD patterns and PDF results allowed us to quantitatively compare the vertical and lateral dimensions as well as the stacking disorder of nanosized δ-MgCl₂ crystallites with the aid of simulation using MgCl₂ nanoparticle models. Moreover, the time-course development of δ-formation by mechanical activation was investigated.

The PXRD patterns of the mechanically activated MgCl₂ samples are summarized in Figure 1. The pristine MgCl₂ exhibited intense and sharp diffraction peaks, which can be assigned as α-MgCl₂ (Figure 1A). Diffraction peaks in all the other series of samples were weakened and broadened along with the grinding time regardless of the absence (B) or presence of additives (C, D, E). In particular, the intense two peaks that relates to the body-diagonal direction, (012) and (104), eventually merged into one broad peak at ~25–40° in 2θ in any series of samples. This indicates the occurrence of the stacking disorder among MgCl₂ crystalline layers [4]. Weakening of the (003) and (110) diffraction peaks corresponds to miniaturization of MgCl₂ crystallites in the c-direction (thinning) and the a,b-directions, respectively [4]. In comparison with the additive free samples (Figure 1B), the presence of EB and DBP evidently accelerated the conversion into δ-MgCl₂ (Figure 1C,D). The acceleration was more obvious for DBP. Note that DBP-3–24 remained pronounced sharp diffraction peaks that can be assigned as (012), (006), (104), and (110) of α-MgCl₂ over the diffuse pattern of δ-MgCl₂. This fact suggests a significant distribution in the crystallite structure. Such inhomogeneity was not observed for the other series of samples. Different consequences were obtained by the addition of toluene (Figure 1E): In Tol-6 and 12, all the diffraction peaks were sharper than those of Dry-6 and 12. This fact suggests that the addition of an inert liquid dissipates the mechanical energy to slow down both the crystallite miniaturization and stacking disorder. Nonetheless, further grinding led to similarly broad diffraction peaks compared to the dry samples (Tol-24,48 vs. Dry-24,48). As an example of chemically activated δ-MgCl₂, a ZNC sample that was prepared from Mg(OEt)₂ using DBP as a donor was also subjected to the analysis (Figure 1F) [4]. Note that its MgCl₂:DBP molar ratio of 15:1 is very close to 16:1 of the DBP-xx samples. It can be seen that the most finely grounded δ-MgCl₂ sample (DBP-48) still remains a more ordered structure compared to the Mg(OEt)₂-based catalyst. To quantify the progress of the δ-MgCl₂ conversion, the experimental PXRD patterns were fit with simulated patterns derived from MgCl₂ nanoparticle structure models [4]. A typical fitting result is shown in Figure S1. The determined structural parameters are summarized together with PDF fitting results and described later.

PDFs that were derived from the total scattering data are shown separately in long-range and short-range regions (Figure 2). The PDF of the Mg(OEt)₂-based catalyst is also given. A PDF includes a lot of important structural information. The peak positions directly correspond to atomic distances; specifically, the first peak at 0.25 nm reflects the closest Mg–Cl atomic pairs, and the second peak at 0.36 nm corresponds to the next closest atomic pairs (Cl–Cl and Mg–Mg pairs) [4]; therefore, the broadness of the peaks relates to the atomic displacement from the ideal positions. Besides, the intensity of the peaks reflects the relative abundance of the atomic pairs, and thus the damping of the peaks along with r is generally related to the finite crystalline dimensions. For δ-MgCl₂, the stacking disorder also causes intensity damping because it spoils the out of plane structural order [4]. In the long-range region (Figure 2A), almost no damping was observed for pristine MgCl₂ up to 10 nm, indicating that its crystallite dimension is sufficiently larger than 10 nm. For Dry-3, the peaks did not disappear at r = 10 nm, but the intensity around 10 nm was relatively smaller compared to the pristine MgCl₂. Judging from the sharpness of (003) and (110), the damping was most probably induced by the stacking disorder. On the basis of this speculation, the probability of α-MgCl₂ sequence (P_c, see Section 3.3) was determined by PDF fitting with MgCl₂ structure models in an infinite particle boundary mode. Note that it is not affordable by PXRD fitting due to the computationally infeasible size of the crystallite. P_c of EB-3 was also determined by the same method. On the other hand, Dry-6, Dry-12, Dry-24, Dry-48, EB-24, DBP-24, and Tol-24 exhibited significant decay, reflecting the fact that their crystallite dimensions are much smaller compared to the former described samples. The average crystallite dimension of those samples was also estimated by PDF fitting as D (Table S1, see Section 3.4 for details) and the results were well consistent with the crystallite dimension derived from the PXRD patterns. In the short-range region (Figure 2B), the peak positions and their broadness remained unchanged from those of pristine MgCl₂ for all the samples. This indicates that the local structure of MgCl₂ is not affected by mechanical or chemical activation as well as the presence of organic additives including donors. The results of PDF fitting also support this conclusion as the lattice constants (l_a,b,c) as fitting parameters were quite similar to each other (cf.

Table 1. PXRD fitting results for mechanically activated MgCl₂ samples.

Sample Name	Pc [%]	La × Lb [nm]	Lc [nm]
Pristine MgCl2	100 1	61.5 2	72.4 2
Dry-3	91 3	56.8 1	54.2 1
Dry-6	58	12.7 × 12.7	13.3
Dry-12	55	12.7 × 12.7	13.3
Dry-24	40	10.9 × 10.9	11.5
Dry-48	40	10.9 × 10.9	11.5
EB-3	88 3	38.7 2	41.5 2
EB-6	58	14.5 × 14.5	13.3
EB-12	58	14.5 × 14.5	13.3
EB-24	55	10.9 × 10.9	10.3
EB-48	46	9.1 × 9.1	8.0
DBP-3	55	12.7 × 12.7	13.3
DBP-6	34	9.1 × 9.1	9.7
DBP-12	37	10.9 × 10.9	8.0
DBP-24	31	9.1 × 3.6	6.2
DBP-48	31	9.1 × 3.6	6.2
Tol-6	82	29.1 × 29.1	29.2
Tol-12	64	16.4 × 16.4	16.2
Tol-24	46	12.7 × 12.7	11.5
Tol-48	31	9.1 × 9.1	8.0
Mg(OEt)2-based catalyst	25	7.6 × 3.3	0.9

¹ Pristine MgCl₂ was confirmed as highly crystalline α-MgCl₂ from the PXRD pattern (Figure 1A). Thus, 100% of P_c was assumed. ² The crystallite dimensions are too large to simulate the PXRD pattern based on the Debye equation. Instead, the Scherrer equation and the full-width at half-maximum (FWHM) of the (003) and (110) diffraction peaks were used to calculate the crystallite dimensions [25]. (D_hkl = 0.94λβ⁻¹cosθ⁻¹, λ: X-ray wavelength, β: FWHM of the diffraction peak, θ: diffraction angle of the peak). ³ P_c was determined by PDF fitting using MgCl₂ structure models in an infinite particle boundary mode. The structure which led to the lowest R_w,PDF value was considered as the best-fit model.

). A typical fitting result and the obtained fitting parameters are shown in Figure S2 and Table S1.

The four structural parameters, P_c, L_c, and L_a,b were determined via PXRD and PDF fitting. They are listed in Table 1, and its graphical summary is provided in Figure 3. As was discussed on the basis of qualitative observation in Figure 1 and Figure 2, the stacking disorder (Figure 3A) as well as the crystallite miniaturization (Figure 3B,C) promoted along with the grinding time. It was found that the crystallite miniaturization underwent in an isotropic manner: There was no significant difference between the values of L_c and L_a,b. This is somewhat interesting in a sense that the repetition along the c-axis is based on the weak van der Waals interaction with respect to the ionic bonding along the a,b-axes. The mechanical force which was isotropically imposed with a sufficient energy to break the ionic bonds plausibly explains the result. Note that δ-MgCl₂ possessed a platelet crystallite morphology when chemically activated. Compared to the dry milled samples, the addition of toluene obviously decelerated the conversion into δ-MgCl₂. Specifically, P_c, L_c, and L_a,b of Dry-6–12 were 58–55%, 13–13 nm, and 13–13 nm, respectively, while those of Tol-6–12 were 82–64%, 29–16 nm, and 29–16 nm, respectively. This could be explained by the physical effects that the presence of a liquid absorbed the collision energy. However, the extent of disorder eventually reached a similar level to that of the dry milled sample at the longest grinding time (P_c, L_c, L_a,b: 40%, 12 nm, and 11 nm for Dry-48, respectively, while 31%, 8 nm, and 9 nm for Tol-48, respectively). Contrary, the addition of donors accelerated the δ-MgCl₂ formation as is shown in the reduction of the three structural parameters at 3 h: P_c, L_a,b, L_c: 91%, 57 nm, 54 nm for Dry-3; 88%, 39 nm, 42 nm for EB-3; and 55%, 13 nm, 13 nm for DBP-3, respectively. In particular, DBP most effectively accelerated the δ-MgCl₂ formation. It is clear that the stabilization of lateral surfaces through chemisorption accelerates the miniaturization along the a,b-axes. On the other hand, the mechanical grinding generally leads to isotropic particles [22,23]. It is considered that the miniaturization along the a,b-axes increases a chance of collision at lateral surfaces and thus in turn promotes the miniaturization along the c-axis. The promoting effect of the adsorption overcomes the retarding role of a liquid additive.

As stated above, the addition of additives exerted significant influences on the progress of the δ-formation. On the other hand, when samples were compared at the longest grinding duration (48 h), only small structural differences were found. More in details, according to the data reported in Table 1 and visualized in Figure 3, both donors slightly promote nanosizing and disorder, DBP a little bit more, while EB at the same level as toluene. Anyhow, it is generally known that the size of particles that can be reached by ball-milling largely depends on other grinding conditions, in particular the size of the balls [24], which plausibly explains the small differences. Finally, the structure of DBP-48 is compared with that of the chemically activated sample in the presence of DBP (Mg(OEt)₂-based catalyst). The P_c and the lateral dimensions, L_a,b were similar (31% and 9 × 4 nm vs. 25% and 8 × 3 nm), while there was a large difference in the c-axis dimension (6 nm vs. 1 nm). This fact suggests that the structure of δ-MgCl₂ largely depends on the preparation protocol.

Finally, FT-IR spectroscopy (in both mid and far IR regions) was performed on the samples at the longest grinding time, in order to assess the possibly different effects of the different mechanical treatments on MgCl₂ morphology. In particular, the investigation focused on the comparison between the dry ball-milling and the ball-milling in the presence of the electron donors (in Figure 4), in order to verify their role as capping agents.

Figure 4A displays the Far IR spectra of the three samples. Such spectra are dominated by a broad absorption centered at 250 cm⁻¹, which is due to Mg–Cl vibrations and we have recently demonstrated to be sensitive to the relative extent of MgCl₂ surfaces [15]. In particular, the spectrum of Dry-48 is the spectrum with a larger absorption at low wavenumbers (~220 cm⁻¹), where the bulk and the basal (001) surface mostly contribute. The spectra of EB-48 and DBP-48 are very similar to each other, slightly differing only at high wavenumbers, indicating a slight difference in the relative extent of the lateral surfaces that give absorption in that region; in particular the spectrum of DBP-48 displays a more pronounced band at 405 cm⁻¹, indicating that DBP more effectively stabilizes the surfaces presenting an absorption there around, i.e., (110), (012), and (015) according to DFT simulations [15].

Afterwards, in order to get a deeper insight on the surface properties of δ-MgCl₂, the IR fingerprints of the donors were analyzed as internal molecular probes, comparing those results with the IR fingerprints of the same donors adsorbed on Dry-48 sample (and outgassed at 90 °C under dynamic vacuum to remove the excess). A curve fitting analysis was applied to the signals in the 1800–1550 cm⁻¹ region, identifying each contribution on the base of the literature.

For what concerns EB (Figure 4(B,B’)), the signals were interpreted as ν(C=O) of EB in interaction with 5-coordinated Mg²⁺ (on (104) surface) at 1700 cm⁻¹, with 4-coordinated Mg²⁺ (on (110), (012), and (015) surfaces) at 1680 cm−¹, and with 3-coordinated Mg²⁺ (defects) at 1650 cm⁻¹, alongside the δ(OH) of a few traces of adventitious water contaminations at 1620 cm⁻¹ and two vibrational modes of the phenyl ring at 1602 and 1583 cm⁻¹ [26,27]. Although a quantitative analysis should have required the extinction coefficients, the relative proportion of the ν(C=O) bands (reported as percentage in Figure 4) reveals that in EB-48 sample there are a little bit more defects than in Dry-48 (32 to 26%) and that the ratio between 4 and 5-coordinated Mg²⁺ sites is slightly higher in EB-48 than in Dry-48 (61:7 with respect to 65:9).

Analogously, also the spectrum of DBP (Figure 4(C,C’)) presents two vibrational modes of the phenyl ring (at 1580 and 1595 cm⁻¹), the δ(OH) of a few water traces and several bands assigned to the ν(C=O) of different DBP molecules. In particular, the band at 1730 cm⁻¹ is due to liquid-like physisorbed DBP, the bands at 1705 and 1680 cm⁻¹ are overall due to DBP on surfaces, and the band at 1660 cm⁻¹ is due to DBP on defects [26,28,29]. In agreement with what observed upon EB analysis, DBP-48 presents more defects than Dry-48 (29 to 18%), whereas in Dry-48 there is a higher amount of liquid-like DBP (33 to 28%), likely because when DBP is adsorbed in a second step the aspecific physisorption is favored. Unfortunately, DBP does not allow distinguishing among MgCl₂ surfaces, since ν(C=O) for DBP on surface Mg²⁺ sites depends not only on Mg²⁺ acidic strength but also on DBP configuration (e.g., monodentate or chelate) [29].

Overall, the FT-IR morphological investigation pointed out that ball-milling in the presence of donors favors the expression of lateral surfaces rather than the basal one and the presence of defects, in well agreement with recent theoretical studies [30,31,32]. However, the differences with Dry-48 sample are really small (as observed in the structural analysis), reasonably because other grinding parameters play a more important role, such as the size of the balls [24].

3. Materials and Methods

3.1. Sample Preparation

All the samples were prepared by mechanically grinding a highly crystalline MgCl₂ powder, which was donated by Toho Titanium Co., Ltd. (Kanagawa, Japan). A 0.5 L stainless-steel pot was charged with 25 g of pristine MgCl₂ and 235 stainless-steel balls (10 mm diameter) under an inert atmosphere. Then, the mixture was ground for 3, 6, 12, 24, or 48 h on a planetary ball mill (Sansho Industry Co., Ltd., Osaka, Japan, a custom-made model). Grinding was performed (i) in a dry condition, and in the presence of (ii) 2.3 mL of ethylbenzoate (EB), (iii) 4.3 mL of di-n-buthylphthalate (DBP), or (iv) 2.3 mL of toluene (Tol). These chemicals were purchased from FUJIFILM Wako Pure Chemical Corporation (Tokyo, Japan) and dried over 4A molecular sieves prior to use. Thus prepared samples are termed as Dry, EB, DBP, Tol-xx: xx is the duration of the grinding in hours. All the samples were handled and stored under an inert atmosphere in order to minimize the moisture contamination.

3.2. X-ray Total Scattering Measurement

The details of the experiment were described in the previous literature [4]. Each powder sample was filled in a Lindemann glass capillary tube (0.3 mm internal diameter). The tube was flame-sealed and subjected to X-ray total scattering experiment at beamline BL05S2 in Aichi Synchrotron Radiation Center (Aichi, Japan). The wavelength of X-ray was set to 0.69 Å (18.0 keV), and the scattered X-ray was collected by four 2D detectors in transmission geometry in the 2θ range of 3–132°. The scattering intensity for an empty capillary tube was subtracted as the background. In order for direct comparison with literature, the diffraction angle of the PXRD patterns was scaled to the wavelength of Cu Kα radiation (λ = 1.5418 Å).

3.3. PXRD Fitting

The acquired PXRD patterns were analyzed based on the comparison with the simulated patterns derived from MgCl₂ nanoparticle models by using DISCUS [33]. The basic concept was already reported in the previous literature [4]. Briefly, nanoparticle models were created based on four parameters: L_a, L_b, L_c, and P_c. The dimension in the lateral directions (L_a,b) is in accordance with unit cells repeated in the a,b-directions. The basal dimension (L_c) corresponds to the number of Cl-Mg-Cl layers stacked along the c-direction. The probability of α-MgCl₂ (ABCA) sequence (P_c) is tuned in 25–100% to express the stacking disorder. The rest 75–0% were equally distributed to β-MgCl₂ (ABAB) and the two rotational error sequences (ABAC, ABCB). The lattice constants were fixed as 0.3636 nm (l_a,l_b) and 0.5889 nm (l_c), which corresponds to the separation of neighboring Cl-Mg-Cl layers. The thickness of a single Cl-Mg-Cl layer was set to 0.2667 nm. These values were taken from experimentally determined lattice constants of α-MgCl₂ [34]. An adequate number of models were generated for a single set of parameters, and their PXRD or PDFs were averaged in order to express stacking probability. The presence of organic additives was disregarded because they composed of only light atoms (H, C, and O) having the very low X-ray scattering power.

Debye scattering equation was used to simulate the PXRD pattern from a MgCl₂ structure model with the aid of DISCUS [33],

$$I_{calc}\left(Q\right)={\displaystyle \sum }_i{\displaystyle \sum }_j{f}_i{f}_j\frac{\sin \left(2\pi Q{r}_{ij}\right)}{2\pi Q{r}_{ij}},$$

(1)

where I(Q) is the scattering intensity at scattering vector Q (4πsinθ/λ), f_i, f_j, and r_ij are, respectively, the X-ray atomic scattering factors for ith and jth atoms, and the distance between the two atoms. This equation inherently depicts the orientation-averaged scattering intensity from a particle model and includes the finite crystallite size effect (peak broadening) [35]. The difference between a simulated and an experimental pattern ((I_calc(2θ)) and I_exp(2θ)) was derived by

$$R_{w,PXRD}={\left[\frac{\sum {\left(I_{exp}\left(2\theta \right)-I_{calc}\left(2\theta \right)\right)}^2}{\sum I_{exp}{\left(2\theta \right)}^2}\right]}^{\frac{1}{2}}.$$

(2)

The four structural parameters were recursively modified to minimize the difference, R_w,PXRD.

3.4. Acquisition and Fitting of PDF

A reduced atomic pair distribution function (G_exp(r)) was converted from the total scattering data using software PDFgetX2 [36]. To acquire the total scattering structure function, S(Q), the 1D scattering data were corrected for X-ray polarization, Compton scattering, and atomic scattering factors. G_exp(r) is obtained by a Fourier transform of S(Q) [17],

$$\begin{array}{ll}{G}_{exp}\left(r\right)& =\left(\frac{\pi }{2}\right){{\displaystyle \int }}_{Q_{min}}^{Q_{max}}\left[S\left(Q\right)-1\right]\sin \left(Qr\right)\mathrm{d}Q\\ & =4\pi r\left[\rho \left(r\right)-{\rho }_0\right]\end{array},$$

(3)

where Q, ρ(r), and ρ₀ are the scattering vector, the atomic pair density distribution, and the average atomic density, respectively. The Q range for the Fourier transform was set as 0.55 to 15 0.55 Å⁻¹ as Q_max and Q_min.

As with the PXRD fitting, the G_exp(r) was compared with simulated G_calc(r) with software PDFgui [37]. G_calc(r) was calculated with a given nanoparticle model by using the following equations,

$$G_{calc}\left(r\right)=\left(G_{calc}^{\prime }\ast S\right)\left(r\right),$$

(4)

$$G_{calc}^{\prime }\left(r\right)=B\left(r\right)\left(\frac{1}{Nr}{\displaystyle \sum }_i{\displaystyle \sum }_j\left[\frac{c_i{c}_j{f}_i{f}_j}{f^2}{T}_{ij}\left(r\right)\right]-4\pi r{\rho }_0\right),$$

(5)

$$T_{ij}\left(r\right)=\frac{1}{\sqrt{2\pi }{\sigma }_{ij}}exp\left[-\frac{{\left(r-r_{ij}\right)}^2}{2{\sigma }_{ij}^2}\right],$$

(6)

$$S\left(r\right)=\frac{sin\left(Q_{max}r\right)}{r},$$

(7)

where N is the number of atoms and c_i is the relative abundance of the ith element. σ_ij is the broadening factor including anisotropic displacement parameters as well as correlation motion of the atoms. These parameters were recursibly updated to reduce the residual R_w,PDF,

$$R_{w,PDF}={\left[\frac{\sum {\left(G_{exp}\left(r_i\right)-G_{calc}\left(r_i\right)\right)}^2}{\sum G_{exp}{\left(r_i\right)}^2}\right]}^{\frac{1}{2}},$$

(8)

B(r) is a damping factor regarding the resolution of instruments. Q_damp was acquired with a standard sample and set as constant. The broadening derived from the limited Q_max value was incorporated by S(r). The finite object effect was taken into account by applying a spherical shape function of Equation (9),

$${\gamma }_{Sphere}\left(r,D\right)=\{\begin{array}{ll}1-\frac{3}{2}\left(\frac{r}{D}\right)+\frac{1}{2}{\left(\frac{r}{D}\right)}^3& \left(r\le D\right)\\ 0& \left(r>D\right)\end{array},$$

(9)

where D expresses the diameter of the spherical object, which is a built-in function of PDFgui. The effectiveness of this approximation was already confirmed for mechanically activated samples [4].

3.5 Acquisition of FT-IR Spectra

FT-IR spectra were acquired in transmission mode with a Bruker Vertex70 instrument, both in the Mid-IR (with an MCT detector) and in the Far-IR (with a DTGS detector). In both cases, the samples were prepared in glove-box as thin layers on a Si wafer (using hexane as a suspending agent), and placed inside a quartz measurement cell connected to a vacuum line to control the atmosphere on the samples during the FT-IR acquisition; the cell was equipped with two KBr windows for Mid-IR spectra, or with two polyethylene windows for Far-IR ones. All the acquired spectra are shown in absorbance, after subtracting the weak contributions of the Si wafer and of the cell windows.

4. Conclusions

In the current study, we attempted to investigate the formation of δ-MgCl₂ by mechanical activation with or without donors. The δ-MgCl₂ samples were prepared with planetary ball-milling in various grinding durations in the absence or presence of donors (ethylbenzoate (EB), di-n-buthylphthalate (DBP), or toluene as a non-coordinative liquid additive). Quantitative structural determination by synchrotron X-ray total scattering revealed that the activation was accelerated by the addition of donors, which was probably caused by stabilization of lateral surfaces of crystallites through chemisorption. However, the structure and morphology of δ-MgCl₂ reachable at a sufficiently long grinding duration were only slightly affected by the additives, in terms of promotion of nano-sizing, disorder and defects. It was concluded that donors accelerate the δ-MgCl₂ formation, but its influence cannot overcome the impact of the activation protocol or conditions.