3.1 Partial radial distribution function and coordination numbers

The partial radial distribution function (RDF) is commonly used to investigate the characteristics of the short-range order. The RDF describes the variation of the probability density as a function of distance from a specified particle. The formula for the RDF is as follows:

where V is the volume of the simulation cell, while $N_i$and $N_j$ are the total numbers of ions i and j, respectively. The term $n_{ij}\left(r-△r/2,r+△r/2\right)$ is the average number of ion j surrounding ion i within the distance of r± r/$2$. The RDF describes the structural ordering of the slag in different ranges, and also provides fundamental structural information including average bond lengths. The RDFs are shown in Fig. 1, while Table 3 shows the average interatomic distances of atoms (which is our main focus and discussed later). In these glasses, the bond lengths for P–O and Al–O were centered at 1.55 and 1.75 Å, respectively, and were similar to those of the other aluminophosphate glasses measured experimentally [31,54]. The average Al–F and P–F bond distances were 1.55 and 1.81 Å, respectively, which is close to that of the other MD simulations [35,36,47], where the Al–F and P–F bond lengths were found to be approximately 1.59 and 1.83 Ǻ.

Fig.1.Full-screen View

(Color online) RDFs of cation-anion: P–O (a), P–F (b), Al–O (c), Al–F (d), Sr–O (e), and Sr–F (f) pairs. In the figures, configurations F0, F15, and F25 correspond to the blue, red, and black lines, respectively.

The coordination number can be defined using the g(r) of the RDF and it can be calculated by integrating the corresponding g(r) curve to the first valley, using the following formula:

The studied average coordination numbers are listed in Table 4, and others are analyzed in detail in the following sections. Considering both the oxygen and fluorine coordination, the coordination number of phosphorus was close to 4 and the coordination number of aluminum was more than 4, due to penta-coordinate and hexa-coordinate Al. The coordination numbers of calcium and sodium (calculated as the sum of the F and O contributions) were close to 6 and 7, respectively. The focused average coordination numbers of strontium were in the range of 5.43–6.10. Obviously, the mean oxygen coordination numbers of metal cations decrease with the addition of fluoride anions, where F competes with O, resulting in an increase in the coordination number of F.

3.2 Local environments of phosphorus

For both compositions containing F, the first peak in the phosphorus–oxygen partial radial distribution function (g(r); Fig. 1a) shows that the P–O distance is 1.55 Å. In an F-free glass (F0), the distance is 1.57 Å, which is in agreement with the observed nearest-neighbor distances reported in other papers [41,55]. Usually, in an F-free glass the P–O coordination number is exactly four, reflecting the tetrahedral structures around the P atoms. In this study, the P–O coordination number was slightly over four, at 4.004. We found lower oxygen coordination numbers: 3.83 for F15 and 3.76 for F25. By examining g_P-F (r) (Fig.1b) and the P–F coordination numbers, 0.17 for F15 and 0.28 for F25, we know that there is an amount of P-F bonding, and that fluorine atoms replace one of the oxygen atoms in the PO₄ units, retaining the tetrahedral structure around the phosphorus atoms at low F concentrations. In general, the phosphorus atoms are essentially four-coordinated, which both oxygen and fluorine are considered at the same time [35]. When P–F bonding occurs, one of the fluorine atoms takes the place of one of the oxygen atoms in the PO₄ unit, generating a PO₃ F tetrahedron. However, at high F concentration (F25), the P–X (O, F) coordination number exceeds four (4.04), indicating that five-coordinate PO₃ F₂ indeed exists in the structural units breaking the structure of PO₄ units. Generally, the aluminophosphate network combines with the corner sharing of the PO₄ and AlO₄ tetrahedral units. The network connectivity (NC) can be used to show the mean number of bridging oxygen atoms in PO₄ tetrahedral units. The presence of PO₃ F₂ breaks the glass structure by reducing the NC such that the glass system becomes unstable. Segmental examples of these structural units are illustrated in Fig. 3.

Fig.3Full-screen View

(Color online) Examples of structural changes, from tetrahedral to hexahedral. The surrounding atoms have been shrunk for clarity (P = magenta, O = red, F = silver, Al = green).

At both F-concentrations, no bridging F atoms (atoms in a P–F–P bridge) were observed, due to the single negative charge of F ions. It is difficult to charge-balance two phosphorus atoms, due to the double negative charge of the oxygen atom [36]. Meanwhile, rare-bridging F atoms appear as P–F–Al in F15 and F25 (Fig. 2h). To investigate the tetrahedral nature of the phosphate units, the O–P–O and P–O–P bond-angle distributions were calculated (Figs. 2a and 2c), showing an O–P–O distribution peak (110°) close to the ideal PO4 tetrahedral angle (109.5°) and a P–O–P distribution peak (near 149°) exceeding the ideal tetrahedral angle (141.2°) for both compositions. The O–P–F bond-angle distributions (Fig. 2f) peaked at slightly lower angles (105°), as the substitution of F atoms by O atoms alters PO₃ F to be slightly different from PO₄ in the glass structure. However, in F25, there was another distribution peak (174°), showing that an F atom was almost in line with an O atom.

Fig.2.Full-screen View

(Color online) Bond-angle distributions for F0 (blue), F15 (red), and F25 (black) for O–P–O (a), O–Al–O (b), P–O–P (c), P–O–Al (d), Al–O–Al (e), O–P–F (f), O–Al–F (g), P–F–Al (h), F–P–F (i), F–Al–F (i), and Al–F–Al (j).

This case is also shown in Fig. 2i, where F–P–F bond-angle appears, indicating that there was a PO₃ F₂ unit in the simulated glass. As the presence of PO₃ F₂ is not supported by empirical evidence, further studies need to be carried out to examine this result.

3.3 Local environments of aluminum

In F0, the Al–O coordination number was 4.008, implying that the majority of the aluminum atoms were in a tetrahedral coordination. In the other compositions the coordination numbers were 3.91 (for F15) and 3.69 (for F25). Nevertheless, the Al–F coordination numbers were 0.50 for F15 and 0.99 for F25. The Al–X (O, F) coordination number exceeded four in both compositions: 4.41 for F15 and 4.68 for F25.

Similarly to silicon in glass, phosphorus and aluminum are both network formers and have tetrahedral coordination with oxygen [55] in F-free glass. In other studies, aluminum atoms were found to be primarily in tetrahedral coordination state in aluminophosphate glasses with increasing alumina contents, whereas octahedrally coordinated aluminum atoms play a major role with a small quantity of Al₂ O₃ contents [56–59]. However, Al–F bonding is slightly different from the P–F bonding, because F atoms not only replace one oxygen atom in an AlO₄ unit, but replace other oxygen atoms as well to create five-coordinate [AlO_x F_y ], leading to more F atoms being included in AlO₄. Similarly, Stamboulis et al. [33] studied the ¹⁹F MAS NMR of calcium fluoro-alumino-silicate glasses and found that as glass it forms [AlO_x F_y ]ⁿ⁻, where x = 3–6, y = 6 − x, and n is the charge of the total complex in Al(IV), Al(V), and Al(VI) coordinate states. We believe that some of the F atoms replace oxygen atoms to form AlO₃ F, while others intrude AlO₄ units and transform them to AlO₄ F in low F concentrations. For the F25 composition, increasing numbers of F atoms bonded to aluminum atoms, creating more non-tetrahedrons such as AlO₃ F₂, AlO₃ F₃, and others. These need to be examined in further F-NMR experiments. Segmental examples of these structural units are illustrated in Fig. 3. A large number of P–O–P bonds are replaced by P–O–Al bonds [60] which play a significant role in increasing the network connectivity and nuclide containment for phosphate-based glasses, enabling their modification for specific applications.

To investigate the tetrahedral nature of the phosphate units, the O–Al–O bond-angle distributions were calculated (Fig. 2b). These showed distribution peaks of 108° for F25 and 105° for F15, which are close to the ideal tetrahedral angle (109.5°). As can be seen in Figs. 2d and 2e, the P–O–Al and Al–O–Al bond-angle distributions were scattered. The O–Al–F bond-angle distributions are shown in Fig. 4b, with a first peak (approximately 85°) that is slightly low. We believe it is possible that the replacement of F atoms and the simulation process do not use three-body potential for O–Al–O. Considering the Al (V) and Al (VI) coordinate states, the angle distribution curve continues up to 180°. Figures 2i and 2j, provide consequential information about the Al–F interaction. At low F concentration (F15), there were no bridging F atoms observed (atoms in an Al–F–Al bridge), and it was the same for the F–Al–F bond-angle distributions. This is different from the F25 composition, where bridging F atoms (atoms in an Al–F–Al bridge) were observed, albeit in small numbers, while the F–Al–F bond-angle distributions showed that at least two F atoms bonded to an Al atom.

Fig.4.Full-screen View

(Color online) Aluminophosphate-rich and Na/Ca/Sr/F-rich regions in aluminophosphate glass (P = magenta, O = red, F = silver, Al = green, Na = purple, Ca = dark blue, Sr = mid blue)

3.4 Fluorine bonding and coordination numbers

In the F15 composition, the F–P coordination number was 0.39, i.e., 2 out of 5 fluorine atoms bonded to a phosphorus atom. In the F25 composition, 17 of the 50 fluorine atoms were bonded, with an F–P coordination number of 0.34. This is in contrast with Christie et al. [35], who calculated much lower F-concentrations. In their study, three of the four fluorine atoms were bonded to a phosphorus atom for F2, and three of the six fluorine atoms were bonded for F6, indicating that the major F atoms impose a constraint in PO₃ F.

As with F–P, the F–Al coordination number was 0.57 in the F15 composition, where 17 of the 30 fluorine atoms were bonded to aluminum. In the F25 composition, 30 of the 50 fluorine atoms were bonded, with an F–P coordination number of 0.59. The fluorine-modifier bonding in both compositions indeed has different characteristics. For the F15 composition, the F–Na bond length is about 2.15 Å, which is slightly shorter than the 2.19 Å of the F25 composition. The F–Na coordination number is 1.04 for F15 and 1.18 for F25. The peaks in the corresponding partial pair distribution functions, g_Sr-F(r) (Fig. 1f) are broader than the typical F–P/Al peak, indicating substantial disorder in the local environments [35]. Similarly, the F–Ca bond length was about 2.21 Å in the F15 composition, which is longer than the 2.15 Å of the F25 composition. The F–Ca coordination numbers were 0.21 for F15 and 0.34 for F25.

As radionuclides in the simulation, Sr atoms are immobilized in the glass, and their release into the environment needs to be avoided. For the F15 composition, the F–Sr bond length was about 2.27 Å, slightly shorter than the 2.29 Å of the F25 composition. The F–Sr coordination number was 0.17 for F15 and 0.60 for F25. The number of F atoms surrounding Sr atoms increases with the increase of the F-concentration.

The F–F coordination numbers (Table 4), were observed to increase with the addition of SrF₂, indicating that the amount of fluorine atoms are increasing around fluorine atoms. The results suggest a tendency of fluorine atoms to concentrate with each other. The coordination numbers of P–Na/Ca and Al–Na/Ca decrease with the addition of SrF₂, which shows that the modified ions tend to deviate from the network former atoms. Doping with fluorine results in the separation of the glass into aluminophosphate-rich and Na/Ca/Sr/F-rich regions, as shown by Fig. 4.

3.5 Simulated atoms and modifier atom bonding

Table 4 shows the Na and Ca coordination numbers, considering both oxygen and fluorine in the first coordination shell. The coordination numbers for Na are in the range of 6.4–6.97, and in the range of 5.7–5.94 for Ca, taking the coordination to both oxygen and fluorine into account. This implies that six and seven are the most common coordination numbers for sodium, and six is the most common coordination number for calcium. The state of strontium is similar to that of calcium, with coordination numbers of 5.43 for F15 and 6.1 for F25. Strontium atoms were observed to be concentrated on aluminophosphate-rich regions by forming 0.06SrF₂·0.98Sr₃ (PO₄)₂·AlPO₄ at low F concentrations.

However, strontium atoms tend to enter into aluminophosphate-rich regions coordinating with more aluminate in the form of 0.8SrF₂ ·1.06Sr₃ (PO4)₂ ·1.9AlPO₄ at high F concentration. This is because strontium competing favorably with calcium/sodium to bond the AlO₄ with the enhancement of strontium atoms. Further studies need to consider the real radionuclide, instead of the non-radionuclide, to achieve a more realistic simulation.

Figure 5 shows the O–M–O, O–M–F, and F–M–F bond angle distributions, where M represents the modifiers of sodium, calcium, and the simulated radionuclide strontium. The O–M–O, O–M–F, and F–M–F distributions follow a similar shape with a very broad distribution of angles (~60–100°). Nevertheless, as the F–M–F distributions correspond to a very small single digit number of bond angles, the F–Ca–F and F–Sr–F distributions merge into a single peak. These distributions indicate a wide range of bonding environments [35], and that the structure of the first coordination shell around the positive valence atoms is octahedral. An anticipated six-coordinated octahedron is expected to show bond angles at 90° and 180°, as the coordination number of Ca is slightly less than six and the coordination number of Na is close to seven, resulting in the slight shrinking of these bond angles.

Fig.5.Full-screen View

(Color online) Bond angle distributions of O–M–O, O–M–F and F–M–F for F15 and F25. M = Na, Ca, and Sr.