Mayer键级

1. Effect of metallic magnesium on enhanced specific heat capacity of chloride molten salts for solar thermal storage applications

• https://doi.org/10.1016/j.solmat.2022.111658

3.1. Bonding characteristics of the Mg/chlorides

For bonding analysis in chemical interactions, the bonding strength and nature can be characterized by a tool of the Mayer bond order [31]. The average Mayer bond orders of the Mg/chlorides in Fig. 3 were calculated from the values of ion pairs within their first coordination shells. The Mayer bond orders of Na+-Cl- and Ca2+-Cl- ion pairs decrease from 0.163 to 0.159 and 0.081 to 0.067 respectively as temperature rises, and the values are nearly 0 for cation-cation and anion-anion pairs in the molten salt. It is consistent with the fact that the bonding interaction between cation-anion ion pairs plays a key role in the molten salt. For the cases with doped-Mg particle, the Mayer bond orders follow an order of Mg–Na+ > Mg–Ca2+ ≫ Mg–Cl-. With the increase of temperature, the corresponding values change from 0.135 to 0.099 and 0.111 to 0.056 for Mg–Na+ and Mg–Ca2+ respectively, while the Mayer bond orders are <0.029 for Mg–Cl-. It can be concluded that the bonding interactions for Mg–Na+ and Mg–Ca2+ ion pairs are obvious, while it can be negligible for Mg–Cl- ion pairs. These results quite agree with calculations based on the ab-initio method of quantum chemistry for molten Li/LiF and Mg/MgCl2 [32]. After Li atoms dissolving into LiF melt, Li atoms react with Li+ ions to form “atomic-cluster-ion” of Limn + type, owing to the delocalization of the valence electron of Li atoms into the vacant orbitals of Li+ ions. Accordingly, it can be inferred that the doped-Mg atoms can react with Na+ and Ca2+ ions instead of Cl− ions in the interface. The valence electron of Mg atoms delocalizes into the vacant orbitals of Na+ and Ca2+ ions, [MgNam]n+ and [MgCax]y + clusters can be formed at the surface of Mg particles.

4. Conclusions

A theoretical study using ab-initio molecular dynamics simulations has been done to investigate the interface effect and enhancement mechanism of the specific heat capacity for the Mg-doped NaCl–CaCl2 molten salt, in terms of the bonding interactions, ion distributions, microstructure evolutions, and potential energies. The Mayer bond orders are 0.135, 0.111 and 0.029 for Mg–Na+, Mg–Ca2+ and Mg–Cl- respectively at 873 K, which proves that there are obvious bonding interactions for Mg–Na+ and Mg–Ca2+ ion pairs, and [MgNam]n+ and [MgCax]y + clusters are formed at the surface of an Mg particle.

2. Electronic structures and physical properties of Na2O doped silicate glass

• https://doi.org/10.1063/1.4987033

The Bond Order (BO) value characterizes the strength and nature of interatomic bonding in a material. It depends not only on the Bond Length (BL) of the bonded atoms but also on the local environment around them. The sum of all BO values gives the total bond order (TBO). When we divide the TBO value by the volume of the cell, the resulting quantity is the total bond order density (TBOD), which is a quantum mechanical metric to represent the strength and internal cohesion of a material. [80,81]

Figure 7(h) shows the calculated TBOD in our models as a function of x. The addition of sodium oxide depolymerizes the silica network and reduces the covalent character of bonding, resulting in the decrease of TBOD. Moreover, the addition of Na2O to silica glass causes a decrease in the Si-O bond strength value, [82] resulting in a decrease in the Total Bond Order Density (TBOD) value with increasing alkali oxide content.

For a more detailed picture of interatomic bonding and the contribution to TBOD from different bonded atomic pairs in our glass models, Figs. 7(a)–7(g) display scattered plots of calculated Bond Order (BO) vs Bond Length (BL) for all pairs. This provides a vivid display of the distributions of bond strength and atomic separations in multi-component glasses, achievable only with ab initio quantum mechanical calculations.

The plots show that Si-O pairs are strongly bonded (covalent bond) with higher BO values than the weakly bonded Na-O pairs (mostly ionic bond). The pie charts in Fig. 7 illustrate the contribution to TBOD from Si-O, Na-O, O-O, Si-Si, and Na-Na pairs. With increasing Na2O content, Si-O contribution decreases while Na-O contribution increases. In high alkali oxide-containing glass (x > 0.4), contributions from Na-Na, Si-Si, and O-O pairs also come into play, but their values are negligibly small.

V. CONCLUSIONS

The compositional dependence of the electronic structures and physical properties of the sodium silicate glasses are successfully explored using the accurate AIMD technique for structural modeling and the ab initio OLCAO method for properties calculation. We have introduced the concept of total bond order density (TBOD) to analyze interatomic bonding and internal cohesion in a complex glass, going beyond traditional geometric analysis. Our simulated glass structures are validated by comparing findings with available experimental results.

The doping of sodium in silica glass disrupts network connectivity, converting bridging oxygen to non-bridging oxygen. The doped sodium ion resides at the interstices created by the network breakup, leading to changes in bond length and bond angle distribution. In the series, mass density and refractive index increase with rising Na content, while TBOD and bandgap decrease nonlinearly. Na and Si ions lose charge to O atoms.

Young’s modulus and shear modulus values decrease with increasing x, while the bulk modulus initially decreases and then increases at x=0.3. The increased Na content makes the glass more ductile.

This study's scope can be extended to investigate the mixed alkali effect and chemical strengthening in silica glass. Similar ab initio modeling can be applied to study alkali silicate glass surfaces and the solvation effect on the glass surface. For such complex systems, the use of ab initio methods and larger models is crucial. The presented results for sodium silicate glass demonstrate the feasibility of such calculations, offering deeper insights into the intriguing ion exchange phenomena of potassium and sodium in sodium aluminosilicate glass. Ongoing studies will explore these aspects further and will be reported at a later time.

3. Stabilization mechanism of arsenic-sulfide slag by density functional theory calculation of arsenic-sulfide clusters

• https://doi.org/10.1016/j.jhazmat.2020.124567

Electronic state of the atoms in the clusters was investigated by Hirshfeld charge (Lu and Chen, 2012b). As shown in Fig. 2b, the Hirshfeld charge of As increases slightly, which reveals that the metallic nature of As gradually increases. As seen in Fig. 2c, more and more S atoms present positive charge with the increase of S-to-As ratio. It suggests that S atoms have a tendency to aggregate around As atoms and form S multimer. This could possibly be explained by the S multimers distributing on the outer shell of the (As2S5)n and the (As2S3)n acting as the core of (As2S5)n. Noticeably, it has been verified that the (As2S5)n is metastable compound. Therefore, based on the analysis of calculation data, the (As2S5)n is most possibly composed of (As2S3)n clusters covered with S dimers on the exterior shell. This is due to the excess S present in the chemical formula of As2S5.

Further bonding information on (As2S5)n clusters were studied by measuring their average length of As‒As, As‒S and S‒S bonds as well as the Mayer bond order (Lu and Chen, 2018). In Fig. 2c, the results show the bond length is irrelevant to the cluster size n, and the order of bond length follows LAs‒As (>2.5 Å) > LAs‒S (2.18 Å) > LS‒S (2.07 Å). Specifically, the Mayer bond order of As‒As is nearly 0, which proves that the As‒As bond has totally broken up. In addition, the Mayer bond order of As‒S and S‒S are nearly 1, which suggests that the strength of S‒S bond is close to the As‒S bond. This indirectly proves the co-presence of S multimers and As‒S clusters in (As2S5)n, which is corresponding to the analysis of Hirshfeld charge and As/S-c distance.

Building on the discussion of the bonding features of various clusters, the increase of S molar ratio would induce the formation of S multimers exteriorly interacting with the As‒S clusters. This sufficiently verifies the above opinion on which the stable structure of (As2S5)n (n = 1–8) is S multimers-covering-(As2S3)n (n = 1–8) structure. The effect of this configuration on the stability of As compound has also been investigated.

4. Quantum Mechanical Metric for Internal Cohesion in Cement Crystals

• https://www.nature.com/articles/srep07332

We find the total bond order density (TBOD) as the ideal overall metric for assessing crystal cohesion of these complex materials and should replace conventional measures such as Ca:Si ratio. A rarely known orthorhombic phase Suolunite is found to have higher cohesion (TBOD) in comparison to Jennite and Tobermorite, which are considered the backbone of hydrated Portland cement.

The bond order (BO) between each pair of atoms represents the quantitative measure of the bond stiffness and strength22. They are important in revealing the origin of internal cohesion in the CSH crystals. We obtain the BO values between every pair of atoms in the crystal using the first-principles orthogonalized linear combination of atomic orbitals (OLCAO) method23 (see Method Section and Supplementary Information for details). Figure 2 shows the BO vs bond length (BL) distributions for four representative CSH crystals from each group (a: alite, b: afwillite, c: suolunite and d: jennite).

As an overall measure of the crystal cohesion, we define the total BO (TBO) to be the sum of individual BO values and the total bond order density (TBOD) as the TBO normalized by the crystal volume.

Figure S3 displays the plots of BO versus BL and the percentage contribution from different types of bonds to the TBO for all 20 crystals. In group a, Belite (a.1) and Alite (a.2), contain only Q0 silicates. The Si-O bonds have a narrow BO distribution centered at ~0.27 with BL centered at ~1.65 Å. The much weaker Ca-O bonds exhibit a larger BO and BL dispersion ranging from 0.02 to 0.10 and 2.25 Å to 3.50 Å, respectively. The Ca-O bonds in Belite have almost evenly spread BO and BL, while those in Alite show some clustering between 2.27 Å to 2.55 Å. These BO and BL distributions clearly explain why the strong and considerably sharper Si-O peaks dominate the experimental vibrational spectroscopic analyses of these minerals in contrast to the broad and weaker peaks associated with Ca-O bonds25,26. It is noteworthy that in Portlandite (a.3) the BO for Ca-O bonds (0.223 with a BL of 2.381 Å) is much stronger than the Ca-O bonds in the above CS crystals and all other CSH crystals.

We propose the total bond order density (TBOD) as an accurate quantum mechanical metric to classify the CSH crystals that provides far deeper insight to cohesion and strength of these materials.

5. Influence of aluminum on structural properties of iron-polyphosphate glasses

• https://doi.org/10.1016/j.ceramint.2020.04.250

Mayer's Bond Order analysis was used to describe bonding mechanisms in the glasses. The existence of the mix-valence iron atoms has been postulated.

In this case, the CN number can be strongly dependent on the assumed cut-off value. Additionally, such an approach gives only information concerning a geometrical arrangement of atoms but not if the atoms form the chemical bonds. To overcome the problem we decided to use Meyer's Bond Order (BO) analysis to describe the glass network short-range order 37,[72(https://www.sciencedirect.com/science/article/pii/S0272884220312074#bib72), 73(https://www.sciencedirect.com/science/article/pii/S0272884220312074#bib73), 74(https://www.sciencedirect.com/science/article/pii/S0272884220312074#bib74), 75(https://www.sciencedirect.com/science/article/pii/S0272884220312074#bib75)] as implemented in the CPMD code.

Exemplary clusters of interacting atoms cut from the simulated glasses with the bond orders and distances to the central atom are presented in Fig. 14. Mean values of BOs, their standard deviations, and bond lengths in different atoms joins are summarized in Table 5.

As one can see, bond orders of atoms in a first coordination sphere are at least of an order higher than in the second. Thus, the parameter is very suitable in the determination of atoms that interact with each other.

In the studied glasses the network species is phosphorous. All of the atoms, for both glasses, are present in tetrahedral coordination (CN = 4) where the first neighbors are only oxygen atoms, as would be expected. The highest bond order about 1.3 is formed with oxygen atoms that are not joined with Fe/Al cations and their bond length is the shortest (c.a 1.47 Å). The oxygen atoms are frequently called as double-bonded due to pentavalency of P. Non-bridging oxygen atoms which create on the other side bonds with Fe/Al are characterized by the lower BO values of about 1.0 and longer bond length (c.a. 1.5 Å). In the case of the iron-phosphate system, there are also present oxygen atoms that take part in the formation of a bond with one P and two Fe atoms. For the oxygens, the P–O bond order is lower (c.a. 0.90) and bond length is about 1.59 Å. The lowest BO parameter and the longest bond lengths are for bridging oxygen atoms which create P–O–P joins. It can be also seen (Table 5) that Al atoms slightly stronger attract oxygens then Fe what results in an elongation of P-ONB bond length and decrease of the bond order. Thus, based on the BOP-O value one can select several ranges of the values depending on the role of oxygen in the glasses. The ranges are about 0.6–0.8, 0.8–1.1, and over 1.1 for P-OB, P-ONBO, Pdouble bondO, respectively. This gives an easy way to determine the numbers of Qi species in a glass. In both simulated glasses there are similar quantities of Q0 and Q2 structural units about 3% and 50%, respectively. In the iron-phosphate glass, there is a higher concentration of Q1 units equal to about 32% whereas the concertation in the aluminum glass is about 22%. This decrease is related to the increase of the Q3 species from 15% to 25% for x = 0 and 30 glass, respectively. It shows the higher glass network polymerization for the aluminum-phosphate than the iron-phosphate glass.

Aluminum oxide is an intermediate glass network component which means that is some conditions can substitute network cations in their positions. Oxygen atoms creating bonds with Al in over c.a. 93% of cases take part in the formation of P–O–Al bridges and only about 7% of the atoms create Al–O–Al joins. In P–O–Al bridges, the bond length is longer and the bond order is smaller than in Al–O–Al (Table 5). The Al–O BO is in the ranges of c.a. 0.4–0.6 and c.a. 0.7–0.9 for P–O–Al and Al–O–Al, respectively. Thus, it can be used as a useful parameter to distinguish the two kinds of joins. For all Al atoms, BO values for the second coordination sphere are an order lower than in the first. In this case, a very important parameter is the CN number. Most of the aluminum atoms are in coordination 4 to oxygen (about 50%) and are placed in the middle of the tetrahedron (Fig. 14b). Little less is occupied coordination 5 (c.a. 45%) in which oxygen atoms are placed in vertices of a trigonal bipyramid. The coordination polyhedron is strongly distorted and looks like tetrahedron with added additional oxygen which creates about 10% longer bond than in the tetrahedrons. Only about 5% of Al is in octahedral coordination (CN = 6). Thus, the mean CN to oxygen is about 4.5. The sum of all bond orders for a given Al atom is in the range of 2.55–2.70 with a mean value of 2.63 (4). This value can be identified with a valency of the atom [72,73] and is close to formal 3 for Al.

More complicated is the iron case. Iron is an intermediate element of a glass network but it has also a possibility to change its redox state and can be present as ferric or ferrous is a glass. As would be expected, the closest neighbors of iron are oxygen atoms and Fe–O bond order is in the range of c.a. 0.120–0.412 with a mean value of about 0.24 (5). The oxygen atoms which take part in the formation of Fe–O bonds in most cases (about 88%) form Fe–O–P joins and the rest (c.a. 12%) create Fe–O–Fe. Among the Fe–O–P joins there is also about 11% of oxygens which take part in a formation of joins with two iron atoms (P–O–2Fe). The most of iron atoms are in coordination 5 to oxygen (about 60%) and are placed in square pyramids (Fig. 14d). The rest of the iron atoms is almost equally populated between tetrahedrally and octahedrally coordinated sites. Another important aspect is the observation of the occurrence of interaction between iron atoms. About 84% of the atoms have in a second coordination sphere one or two other Fe atoms with which they create bonds. Both positions are almost equally populated and all the irons connected with 2 others form 3 or 4 membered Fe rings. The mean bond order in the case of Fe–Fe interaction is about 0.252 with a mean distance of 4.153 Å. The interaction of that kind was observed previously in Ref. [37] for aluminum-iron-phosphate glass and was responsible for iron atoms clustering in the glass. Calculated valency of Fe atoms changes in a continuous broad range from 1.7 to 2.57. The vales nearby the range limits suggest that there are Fe(II) and Fe(III) cations in the glass. But there is also iron of mixed valency Fe(II/III). To evaluate a fraction of iron atoms in the specific redox state we propose to use valency values for oxygen. The calculated valency for oxygen is in the range from 1.52 to 1.97 with a mean value of about 1.843. Therefore iron atoms for which valency is in the above range will be designated as Fe(II), the atoms of valency above 2.28 will be called Fe(III), and the values between those limits will suggest the mixed valency iron Fe(II/III). Thus, in the studied glass, there is c.a. 29% of Fe(II), 30% of Fe(III), and 41% of Fe(II/III). However, it should be notated that the values are only approximated and the change in the limits may produce different fractions.

6.Structural features of 19Al2O3-19Fe2O3-62P2O5 glass from a theoretical and experimental point of view

• https://doi.org/10.1016/j.jnoncrysol.2019.119499

It was observed linear dependence of Mayer's bond order parameters on bond lengths and values of full valency were calculated.

The above clusters were used to calculate diatomic bond orders based on the method proposed by Mayer (MBO) [78]. The variation of the MBO with the bond length between metal cation and oxygen is presented in Fig. 12.

The MBO decreases with the increase of the bond length in the considered bond lengths range. The shortest bond lengths are for Psingle bondONB for which the MBO index is approximately 1.5. The Psingle bondOB bonds are longer and the same index is about 1.0 and are consistent with the values obtained in [79]. Then there is a wide region of Alsingle bondO bonds which orders change in the range of 1.1 to 0.4 depending on the Al coordination number to oxygen. The shorter bond lengths and the higher bond orders are characteristic for Al in tetrahedral coordination than in octahedral. The similar trend is also observed in the case of iron. The Fesingle bondO bond orders are lower and the lowest is characteristic for Fe(II)single bondO for which the bond lengths are the highest. It is worth noting that independent of iron oxidation state and coordination number the MBO indexes form a continuous set. The similar effect of variation with a metal-oxygen bond length of bond orders was previously reported in [80].

The electron densities at iron nuclei were calculated and theoretical Mössbauer isomer shift values were estimated. The obtained theoretical results were used to assigned subspectrum components to proper iron coordination and oxidation state. It was observed linear dependence of Mayer's bond order parameters on bond lengths and bonding properties were presented. The most covalent bond is Psingle bondO, Alsingle bondO then Fe(III)single bondO whereas Fe(II)single bondO bond is mostly ionic.

7. Alternative insight into aluminium-phosphate glass network from ab initio molecular dynamics simulations

• https://doi.org/10.1016/j.ceramint.2020.09.018

In the paper, Mayer's Bond Order analysis was employed as a suitable tool to observe the changes in the network due to Al2O3 substitutions.

The atoms building the network are going to take specific values of the total bond orders and the glass network is an interplay between atoms’ affinity to saturate bond orders and the glass network neutrality.

Additionally, we want to show the possibility and usefulness of bond order analysis in description of glass network structure.

3.2. Mayer's bond order analysis

One of the important parameters which can be used to describe the glass network properties and their influence on the glass properties is the character of chemical bonds. To describe the bonding properties diatomic bond orders based on the method proposed by Mayer (MBO) [71,72] were calculated as implemented in the CPMD code. The variation of the MBO with the bond length between cations and oxygen is presented in Fig. 4.

It can be seen that the bond order decreases exponentially as the bond length increases, and follows the simple formula MBÕ exp (-(d-d0)/b) (fitted line in Fig. 4) proposed by Pauling [73]. The d0 and b are parameters which are empirically fitted. In case of the studied glasses, the values are: for P–O d0 = 1.453 (1) Å, b = 0.141 (3) Å, and for Al–O d0 = 1.449 (22) Å, b = 0.402 (9) Å. The MBOs for P–O take characteristic values depending on the role of oxygen in the glass network. The shortest bond lengths are characteristic for terminal oxygen atoms which formally should create a double bond with phosphorous (Pdouble bondO). For the Ot atoms, the MBOs are in the range of 1.49–1.21 with a mean value of 1.34. The P-ONB bond is longer what results in the lower MBOs in the range of 1.19–0.90 with an average value of 0.99. The longest bond and the lowest bond orders are for P-OB in the range of 0.90–0.40 with a mean value of 0.71. The mean value of the MBOs for Al–O is equal to about 0.55 (4). The MBO values are similar to the obtained previously by different calculation methods [29,48,74,75]. The above ranges of the bond orders for P–O bonds can be used as a useful tool for recognition of the type of oxygen atoms bonded to phosphorous in the glass.

The values of the bond orders are the highest for the interacting atoms in the first coordination sphere. The interactions between the atoms in the further spheres are about an order lower. Exemplary first coordination spheres of P and Al, bond orders and lengths are presented in Fig. 5.

In the case of the [PO4] tetrahedrons the highest bond orders and in this way the strongest interaction is between P and O atoms. Nevertheless, it can be seen much weaker of about an order lower covalent interaction between O–O atoms in the tetrahedron. This interaction is stronger between Ot-OB than between OB-OB. This is the effect that the OB oxygen is common oxygen with other [PO4] tetrahedron and it also interacts with oxygens in the second unit.

The profiles were calculated across the line connecting two neighbouring oxygen atoms. The green line represents the profile between oxygen atoms in [AlO4] tetrahedron. For the atoms, the distance between them is the longest, and the profile has a minimum in the middle. The value of the minimum is almost equal to 0. Thus, the atoms do not interact with each other. This is following the bond order analysis where the MBO is 0. In the case of the bridging oxygen atoms (OB) in Q3 structural units, it can be seen that the distance between the atoms is the shortest and in the minimum the ELF is about 0.3. The minimum for ONB atoms in Q0 structural units is slightly lower and is about 0.35.

Mayer's Bond Order can be also a useful parameter to determine the coordination sphere order for a given ion. As was shown above, the value of MBO is at least an order lower for the ions in the further spheres. For the Al–O pair, the MBO values are in the range of 0.2–0.8 for the oxygen atoms in the first coordination sphere whereas for the others are approximately 0. Thus, the first coordination sphere can be easily determined. This can be important in case of a glass network modifiers where the proper determination of the cut-off radius can be difficult.

Analysing the bond order results we observed some regularities, concerning [PO4] and [AlOx] polyhedrons. It can be seen, that sum of all P–O bond orders inside the tetrahedron is almost constant and equal to 3.47. There are three possible values of P–O bond orders which depend on the role of the oxygen in the network. The most stable is the P-OB bond order value of c. a. 0.71. The P-ONB bond order depends on the Al coordination which takes part in the formation of the P-ONB-Al bridge. The smallest in the range of 0.8–0.9 is in the case where the oxygen is connected to [AlO4] tetrahedrons and the highest for octahedrally coordinated Al in the range of 1.0–1.2. Whereas, for [AlO5] takes values between these two ranges. Finally, for P-Ot is about 1.34. Thus, e.g. in a Q3 unit, there are three OB atoms and one Ot what results in a total MBO of P–O bonds equal 3.47. Similarly, in the case of [AlOx] polyhedrons where the sum of Al–O bond orders is about 2.45 and it does not depend on Al coordination to oxygen. Depending on the Al coordination, the Al–O bond orders are c. a. 0.6, 0.5, 0.45 for Al in coordination 4, 5, and 6, respectively. Moreover, there is a very strong oxygen tendency to obtain total bond order of 1.84. Thus, to fulfil the above rules several characteristic, for a specific glass composition, types of atoms joining are possible, as presented in Fig. 12.

As a basis of our discussion, we decided to use Mayer's bond order analysis which proved to be a useful tool in the glass sanetwork description in place of the simple geometrical considerations based on cut-off radius. The analysis gives the possibility to consider only interacting atoms in the coordination sphere. Thus, it can be especially useful in case of modifier atoms where proper determination of coordination sphere radius can be difficult. The analysis together with the electron localization function showed the covalent interaction between oxygen atoms in the phosphate network. The interaction spreads across the whole phosphate part of the network.

8. Ab-initio molecular dynamics calculation on microstructures and thermophysical properties of NaCl–CaCl2–MgCl2 for concentrating solar power

• https://doi.org/10.1016/j.solmat.2020.110696

3.1. Bonding characteristics of the ternary chlorides

The Mayer bond order is useful in bonding analysis using semi-empirical computational methods, which can obtain the strength and nature of the bonding characteristics [38]. The averaged bond lengths and corresponding Mayer bond orders of the ternary chloride molten salt are shown in Fig. 2. And the average values were calculated from the distances and Mayer bond orders of ion pairs within their first coordination shells. The Mayer bond orders of ion pairs decrease with the increase of averaged bond lengths, while the Mayer bond orders of cation-cation ion pairs are nearly 0. The Mg2+-Cl- pair has the minimum bond lengths of 2.51–2.66 Å and the corresponding Mayer bond orders of 0.19–0.13. The interaction between cation-chlorine ion pairs is much higher than that of cation-cation ion pairs, and there is almost no bonding between cations. It indicates that local coordination structures in the NaCl–CaCl2–MgCl2 ternary molten salt are formed by -Cl--M-Cl-- (M = Na+, Ca2+ or Mg2+) bridges.