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Investigation of multicomponent silicate glasses by coupling WAXS and molecular dynamics
Journal of Non-Crystalline Solids 293±295 (2001) 290±296
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Investigation of multicomponent silicate glasses by coupling WAXS and molecular dynamics J.-M. Delaye a,*, L. Cormier b, D. Ghaleb a, G. Calas b b
a CEA/DCC Service de Con®nement des D echets, CEA Valrh^ o-Marcoule, BP 171, 30207 Bagnols/C eze cedex, France Laboratoire de Min eralogie-Cristallographie, Universit es de Paris 6 et 7 et Institut de Physique du Globe et UMR CNRS 7590, 4 Place Jussieu, 75252 Paris cedex 05, France
Abstract An innovative method coupling molecular dynamics (MD) and reverse Monte Carlo (RMC) analysis allowed us to ®t the interaction potentials on the experimental structure factors of aluminoborosilicate glass compositions obtained by wide-angle X-ray diraction spectrometry (WAXS). By calculating the structural factors and the partial radial distribution functions, the combination of RMC and MD directly identi®es the pair potentials requiring optimization. We observed that a Coulomb potential alone is not enough to model a suciently ordered local environment around the Na and Ca cations: additional terms were necessary to ®t the repulsion and attraction terms. Analysis of the glass atomic structure revealed a larger number of F1±O±F2 groups in the glass containing CaO (where Fi Al or B ions), although the total number of these groups remains lower than would be expected from a random ion distribution. The relatively higher Ca coordination number around Al compared with B ions shows a preferential Ca±Al approach compared with Ca±B. Finally, analysis of the Na and Ca environments showed that the Na ions more readily act as pure charge compensators ± i.e., distant from any non-bridging oxygen atoms ± than do the Ca ions, most of which conserve non-bridging oxygen atoms in their local environment. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 61.43Bn; 61.43Fs
1. Introduction The structure of multicomponent silicate oxide glasses is dicult to investigate because the techniques generally used for ordered structures are unsuitable for vitreous matter. Although the ®rst can be studied by co-ordination shell (1±2 A) various techniques (XAS, NMR, IR, Raman, etc.), is subject to the medium-range order (P5 A) considerable uncertainty, even if recent experi-
* Corresponding author. Tel.: +33-4 66 79 17 94; fax: +33-4 66 79 66 20. E-mail address: [email protected] (J.-M. Delaye).
ments [1] have suggested the presence of mediumscale organized structures. Simulation techniques provide an alternative to experimental methods for investigating the medium-scale order, notably molecular dynamics (MD), which is capable of reconstituting cells of a few cubic nanometers containing thousands (or millions) of atoms [2,3]. One of the essential aspects of simulation methodologies is the validity of the potentials used to describe atomic interactions: the potentials must reproduce the co-existence of covalent and ionic interactions in multicomponent glasses. This work adopted a novel approach coupling wide-angle X-ray diraction (WAXS) and MD
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 6 8 0 - 9
J.-M. Delaye et al. / Journal of Non-Crystalline Solids 293±295 (2001) 290±296
simulation with an intermediate reverse Monte Carlo (RMC) step [4] to ®t the interaction potentials on the experimental measurement data. The method has already been presented [5] without the potential ®tting step described here. An investigation of the relative interactions between cations (Na and Ca) and trivalent network formers (B and Al) is also proposed. 2. Methods 2.1. Molecular dynamics simulation and calculation of structure factors Interactions in vitreous oxide structures are classically represented in MD by Born±Mayer± Huggins potentials associating pair terms (Eq. (1) is the conventional formula to which the authors have added the 1=r6 and 1=r8 terms) and threebody angular terms (Eq. (2)) representing Coulomb interactions and covalent interactions [6,7]: ! rij qi qj C B U2 rij A exp ; 1 r6 r8 qij rij c c U3 rij ; rik ; hjik k exp rij rc rik rc cos hjik
2
cos h0 ;
2
where rij and rik are the interatomic distances, qi and qj the charges, and hjik the angle formed by an atom triplet j±i±k; the other parameters are adjustable parameters depending on the nature of the atoms involved. The Coulomb interactions are calculated using formal charges. The long-range component of these interactions is taken into account by the complete Ewald sum. An initial version of the Na± O and Ca±O potentials with zero B and C parameters was corrected by introducing 1=r6 and 1=r8 terms to obtain a better ®t of the depth of the potential well with the experimental data. The other pair potentials contain no 1=r6 or 1=r8 terms. The glass structures (Table 1) were prepared from a random con®guration. The liquid was stabilized between 4000 and 6000 K, before quenching at a rate of 1015 K s 1 to 1400 K, then
291
Table 1 Glass compositions (mol%) Glass 1 Glass 2
SiO2
B2 O3
Na2 O
ZrO2
Al2 O3
CaO
64.13 60.12
16.81 16.0
13.27 12.64
1.78 1.7
4.01 3.82
± 5.72
at 4 1014 K s 1 from 1400 K to room temperature. The atom positions were averaged over 3000 time steps to calculate the structure factor and for structure analysis. The structure factor was calculated from Eqs. (3) and (4): X S Q Wab QSab Q; 3 a;b
Wab Q;E
ca cb R fa Q;Efb Q;E jhf Q;Eij
2 dab ; 4
where S Q is the total structure factor, Sab Q the partial structure factors and Wab Q weighting factors depending on the atomic fraction (ci ) and on the atomic diraction factors (fi ) of species a and b. 2.2. Glass preparation and WAXS measurements The glasses in Table 1 were prepared by mixing reagent grade oxides. The powders were melted at 1100°C in graphite crucibles and quenched by partial immersion of the crucible. The glasses were then annealed for 1 h at 520 °C. Glass samples were cut as slabs 3 cm in diameter and 4 mm thick. WAXS measurements were performed on a diffractometer equipped with Mo Ka radiation and a bent graphite monochromator (Philips, PW 1729). The intensity measurements were carried out by the h=2h step scanning method with a constant step of 0.25° (2h) and in the angular range 0:5° < 2h < 140°, which corresponds to a Q range 1 (Q 4p sin h=k is the scatfrom 0.8 to 16:6 A tering factor, 2h is the scattering angle and k is the radiation wavelength). The total count accumulated at each measurement point was not less than 20 000. The X-ray source was operated at a current of 35 mA with an accelerating voltage of 50 kV. After correction (for polarization and absorption factors) and normalization, the coherently scattered intensity per atom was used to calculate the total structure factor S Q.
292
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2.3. Potential ®tting method A two-step procedure was adopted in which the potentials were ®tted to the experimental results, then the glass structures were analyzed. Knowing the atomic positions from the MD model, the structure factors, S Q, for the simulation are calculated according to the Eqs. (3) and (4). The MD models are then used as the starting con®guration in a RMC modeling of the experimental structure factors. The partial pair distribution functions calculated from the MD and the RMC models are compared, which allow the determination of the interatomic potentials to modify. The RMC technique has already been used to ®t the atomic structure to experimental spectra [1]. It consists in minimizing the mean square deviation between the experimental and simulated structure factors by a series of random atom dis-
placements. The minimum approach distances between two types of atoms are imposed, and constraints are placed on Si and Al atoms to conserve the tetrahedral environments (the Al tetrahedral co-ordination in these glasses was recently demonstrated experimentally in [8]). There are no constraints for the other co-ordinations. The size of the cell is taken equal to that determined by the MD simulations. Fig. 1 compares the experimental structure factors with the values simulated by MD and after RMC correction. The deviations ± notably the initial double peak on the simulated structure factors ± disappear after the atomic displacements generated by the RMC technique. Fig. 2 shows the partial radial distribution functions for Na±O and Ca±O in glass 2 (the more complex one) as calculated by MD and re®ned by the RMC method. The structure corrected by RMC exhibits much better de®ned local orders
Fig. 1. Experimental structure factors compared with MD and RMC results.
Fig. 2. Partial Na±O and Ca±O radial distribution functions calculated by MD before potential ®tting, compared with the values for the RMC-corrected structure. The curves are shifted for convenience.
J.-M. Delaye et al. / Journal of Non-Crystalline Solids 293±295 (2001) 290±296
Fig. 3. Na±O and Ca±O potentials before and after ®tting (energy in 10
10
293
cgs).
Fig. 4. Comparison of experimental structure factor for glass 2 with ®tted MD values and with RMC values.
(higher intensity peaks) around the Na and Ca atoms, and the ®rst-neighbor distances are shorter than predicted by the initial MD model. We therefore attempted to deepen the Na±O and Ca± O potential wells by adding 1=r6 and 1=r8 terms and reducing the ®rst-neighbor distances. The curves for the Na±O and Ca±O potential pairs are plotted in Fig. 3 before and after ®tting the additional terms; with the deeper potential wells the simulated local orders are better de®ned around the Na and Ca atoms. Other changes included deleting the three-body term for the O±B±O triplets (necessary to better represent the boron coordination recently determined by 11 B NMR measurement) and slightly extending the Si±O ®rst-neighbor distance by increasing the Si±O repulsion. We recalculated the structure factors of the 6-oxide glass by MD using the readjusted potentials, and obtained a signi®cantly better agreement
with the experimental ®ndings (Fig. 4): the ®rst structure factor peak is no longer separated into two smaller peaks, and largely corresponds to the experimental peak. RMC was again used to re®ne the results: after RMC correction (Fig. 4) the simulated and experimental structure factors fully coincide. 3. Results The structures of 5- and 6-oxide glass compositions prepared by MD with ®tted potentials were analyzed, with particular attention to the interactions between the trivalent network formers (Al and B) incorporated in the polymerized network and cations (Na and Ca). The two compositions containing only Na2 O for the ®rst and a mixing Na2 O CaO for the second allow to compare the individual behaviors of Na and Ca. Table 2
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Table 2 Number of F1±O±F2 groups (where F Al or B) in glasses 1 and 2 (5184 atoms) Glass 1 Glass 2
Al±O±Al
Al±O±B
B±O±B
3
8 13
93 122
247 243
42 23
Table 3 Coordination numbers of network formers Al and B with Na radius and Ca within a 4.2 A Glass 1 Glass 2
Al(Na)
Al(Ca)
B(Na)
B(Ca)
1.94 1.9
± 0.57
2.43 2.46
± 0.41
indicates the number of Al±O±Al, B±O±B and Al± O±B groups in the glasses, considering only the oxygen atoms with two neighboring formers. Table 3 lists the Na and Ca co-ordination numbers around the Al and B atoms: Al(Na), for example, corresponds to the number of Na atoms within a spherical radius around the Al. In the fol4.2 A lowing text, the letter F will represent the formers Al or B, and M the modi®ers Na or Ca. The 4.2 A radius corresponds approximately to the minimum following the ®rst peak of the F±M partial radial distribution functions. Adding CaO to the 6-oxide glass had practically no eect on the F(Na) co-ordination number, although the Al(Ca)/Al(Na) ratio was nearly twice the B(Ca)/B(Na) ratio. This observation suggests that Ca tends to be situated near Al atoms rather than near B atoms. The number of F1±2 O±F2 groups that would be obtained in a random distribution can be estimated as follows (2 O signi®es that only oxygen atoms with two and only two neighboring formers are taken into account). Let NO2 represent the number of oxygen atoms with two neighboring formers (which also corresponds to the total number of groups as de®ned here), NF the total number of formers (including Si, B, Al and Zr) multiplied by their average co-ordinations, NF1
B±O±3 B
3
B±O±4 B
136 123
4
B±O±4 B
69 97
the number of type F1 formers multiplied by the F1 co-ordination, and NF2 the number of type F2 formers multiplied by the F2 co-ordination. If F1 and F2 are identical species, a random distribution of network formers would result in a number of F1±O±F2 triplets equal to NO2
NF1 NF1 1 : NF NF 1
5
If F1 and F2 are dierent species, this equation becomes NO2
NF1 NF2 2: NF NF 1
6
Table 4 indicates the number of groups corresponding to a random distribution. In the 5-oxide glass, the number of Al±2 O±Al and Al±2 O±B groups is well below the value expected from a random distribution of network formers. The exclusion principle known in the case of Al as the Lowenstein exclusion rule [9,10] is con®rmed in this glass, and is also applicable to Al±2 O±B triplets. The number of these groups increases with the addition of CaO, but remains below the result expected from a random distribution. Table 4 contains the number of 4 B±O±4 B, 3 B±O±3 B and 3 B±O±4 B groups in the simulated structures assuming a random distribution of 3 B and 4 B (representing 3- and 4-coordinate boron, respectively). The 4 B±O±4 B groups, under represented in the 5-oxide glass, are found slightly under the `normal' amount in the 6-oxide glass. Parallely the 3 B±O±4 B groups are favored in both compositions.
Table 4 Number of F1±O±F2 groups in glasses 1 and 2 assuming a random distribution of B and Al around oxygen atoms Glass 1 Glass 2
Al±O±Al
Al±O±B
B±O±B
3
B±O±3 B
15.7 17
125.3 128.2
248.8 241
34.5 26.5
3
B±O±4 B
116.4 107
4
B±O±4 B
97.9 107.6
J.-M. Delaye et al. / Journal of Non-Crystalline Solids 293±295 (2001) 290±296
4. Discussion We have proposed a novel methodology for ®tting MD potentials to experimental ®ndings. The RMC technique directly reveals the pair potentials requiring correction. It is important to note that in the RMC results the experimental resolution is included. This explains partly the broadening of the peaks in the RMC ®ts compared to the initial MD model. The main dierence concerned the local environments of the Na and Ca cations. It was necessary not only to shorten the ®rst-neighbor distance, but also to order the shell of oxygen atoms around the cations by deepening the pair potential well. Using only Coulomb attraction and exponential repulsion terms alone was not sucient to obtain a satisfactory ®t, hence the additional 1=r6 and 1=r8 terms. This work involves only the atomic structure of the glasses, which we ®t to the experimental structure factors. We did not verify the quality of the potentials for representing elastic or dynamic quantities. The satisfactory reproduction of the experimental structure means that the relative intensities of the atomic interactions are correct, although some deviations may remain with respect to the actual intensities. In particular, hardening the Na±O and Ca±O interactions allowed us to limit the deviation with the intensity of the formeroxygen interactions. The need for this correction probably arises from the use of formal charges. The Coulomb component of the former-oxygen interactions is overestimated because the interactions are considered as purely ionic, whereas the experimental results and electron structure calculations indicate lower estimated charge values [11±13]: for example, ab initio calculations for Si and O clusters [14] yielded charges on the order of 1.6 for Si and about )0.8 for O. An ionic representation with formal charges in addition to the angular term intended to represent the rigidity of the covalent bonds, thus overestimates the former-oxygen interactions compared with the modi®er-oxygen interactions. The corrections implemented to deepen the Na±O and Ca±O potential wells thus compensate for the overestimated former-oxygen interactions.
295
Separating the repulsive eect into an exponential term and 1=r8 term tends to harden the short-range repulsion, while the 1=r6 term hardens the attractive eect. The combination of the two eects creates a deeper potential well. For the moment we consider these extra terms to be only a mathematical tool for re®ning the potentials, with no special signi®cance attached to the multipolar interactions the terms might contain. Fitting the new attraction terms qM with a pure Coulomb qO =r C=r6 from 2 to 3 A term (qeff qO =r) nearly doubles the eective charges qeff on the Na and Ca cations, resulting in an order-of-magnitude agreement between the formal Si charge (+4) and the actual charge of about +1.6. The relative interactions between Al, B, Ca and Na were elucidated by analysis of the atomic structures. The Ca ions tend to approach Al, rather than B atoms. The interaction potentials used do not take into account chemical interactions between elements. Space limitations may account for the preference of Ca ions for the vicinity of Al atoms: the AlO4 groups are larger than the BO4 groups (the ®rst-neighbor distance Al±O is around while the B±O distance is around 1.4 A). 1.75 A Adding CaO also increases the number of Al±2 O±Al, Al±2 O±B and 4 B±2 O±4 B groups (see the de®nition above). The Lowenstein exclusion law appears less applicable in the glass containing CaO than in the 5-oxide glass, although the number of groups remains below the values predicted for a random distribution. These results are consistent with ab initio calculations that show the easier formation of Al±O±Al groups when Ca is present [15]. The double charge of the Ca ions thus favors a closer approach to the trivalent formers in tetrahedral coordination. However, the number of B±O±B groups in both glasses is practically equal to the value expected for a random distribution. This result shows no segregation of boron but is not consistent with recent NMR analyses showing 3 B mixed with the silicate network [16]. The very fast quenching rates applied by MD are poorly suited to a study long-range diusion processes than can be responsible for this phenomenon. Finally, consider the role of Na and Ca as charge compensators or modi®ers. We designated
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as charge compensators all the Na or Ca ions with rano non-bridging oxygen atoms within a 3.4 A dius. In the 5-oxide glass, 57% of the Na ions were charge compensators, compared with 47% of the Na ions and 16% of the Ca ions in the 6-oxide glass. The Ca ions were always closer to the nonbridging oxygens than the Na ions, as con®rmed by the non-bridging oxygen co-ordination numbers around the Na (0.72) and Ca (1.75) in the 6oxide glass. It seems rather paradoxical that adding CaO favors the formation of F1±2 O±F2 groups, whereas the percentage of charge-compensating Ca ions with no nearby non-bridging oxygen atoms remains low. The explanation probably lies in the double charge of the Ca ions, with more ambiguous eects than the single-charged Na ions: the Ca ions can approach non-bridging oxygen atoms without losing their charge-compensating role. Their greater volume probably prevents them from being incorporated in large quantities in the polymerized network as pure charge compensators.
5. Conclusion An innovative method coupling MD and RMC analysis allowed us to ®t the interaction potentials on the experimental structure factors of aluminoborosilicate glass compositions obtained by WAXS. By calculating the structure factors and the partial radial distribution functions, the combination of RMC and MD directly identi®es the pair potentials requiring optimization. We observed that a Coulomb potential alone is not enough to model a suciently ordered local environment around the Na and Ca cations: additional terms were necessary to ®t the repulsion and attraction terms. Analysis of the glass atomic structure revealed that with only Na2 O as modi®ers in the composition, the Loewenstein rule is veri®ed not only for Al±O±Al groups but also for Al±O±B and
4
B±O±4 B groups. With Na2 O CaO, a larger number of Al±O±Al, Al±O±B and 4 B±O±4 B groups are calculated, although the total number of these groups remains lower than would be expected from a random ion distribution. The relatively higher Ca coordination number around Al compared with B ions shows a preferential Ca±Al approach compared with Ca±B. Finally, analysis of the Na and Ca environments showed that Na ions more readily act as pure charge compensators ± i.e., distant from any non-bridging oxygen atoms ± than do the Ca ions, most of which conserve non-bridging oxygen atoms in their local environment. Their role as network modi®ers and charge compensators is less clearly de®ned than for the Na ions. References [1] L. Cormier, P.H. Gaskell, G. Calas, A.K. Soper, Phys. Rev. B 58 (1998) 11322. [2] J.-M. Delaye, D. Ghaleb, J. Non-Cryst. Solids 195 (1996) 239. [3] J.-M. Delaye, V. Louis-Achille, D. Ghaleb, J. Non-Cryst. Solids 210 (1997) 232. [4] R.L. McGreevy, Nucl. Instrum. and Meth. A 354 (1995) 1. [5] L. Cormier, D. Ghaleb, J.-M. Delaye, G. Calas, Phys. Rev. B 61 (2000) 14495. [6] T.F. Soules, J. Non-Cryst. Solids 49 (1982) 29. [7] F.H. Stillinger, T.A. Weber, Phys. Rev. B 43 (1991) 1194. [8] B. Boizot, G. Petite, D. Ghaleb, N. Pellerin, F. Fayon, B. Reynard, G. Calas, Nucl. Instrum. and Meth. B 166&167 (2000) 502. [9] S. Wang, J.F. Stebbins, J. Am. Ceram. Soc. 82 (1999) 1519. [10] W. Loewenstein, Am. Mineral. 39 (1954) 92. [11] C.H. Hsieh, H. Jain, A.C. Miller, E.I. Kamitsos, J. NonCryst. Solids 168 (1994) 247. [12] A.C. Lasaga, G.V. Gibbs, Phys. Chem. Mineral. 14 (1987) 107. [13] K. De Boer, A.P.J. Jansen, R.A. Van Santen, Chem. Phys. Lett. 223 (1994) 46. [14] V.S. Yushchenko, E.D. Shchukin, M. Hotokka, J. Mater. Sci. 29 (1994) 3038. [15] J.A. Tossell, G. Saghi-Szabo, Geochim. Cosmochim. Acta 61 (1997) 1171. [16] S. Wang, J.F. Stebbins, J. Non-Cryst. Solids 231 (1998) 286.
www.elsevier.com/locate/jnoncrysol
Investigation of multicomponent silicate glasses by coupling WAXS and molecular dynamics J.-M. Delaye a,*, L. Cormier b, D. Ghaleb a, G. Calas b b
a CEA/DCC Service de Con®nement des D echets, CEA Valrh^ o-Marcoule, BP 171, 30207 Bagnols/C eze cedex, France Laboratoire de Min eralogie-Cristallographie, Universit es de Paris 6 et 7 et Institut de Physique du Globe et UMR CNRS 7590, 4 Place Jussieu, 75252 Paris cedex 05, France
Abstract An innovative method coupling molecular dynamics (MD) and reverse Monte Carlo (RMC) analysis allowed us to ®t the interaction potentials on the experimental structure factors of aluminoborosilicate glass compositions obtained by wide-angle X-ray diraction spectrometry (WAXS). By calculating the structural factors and the partial radial distribution functions, the combination of RMC and MD directly identi®es the pair potentials requiring optimization. We observed that a Coulomb potential alone is not enough to model a suciently ordered local environment around the Na and Ca cations: additional terms were necessary to ®t the repulsion and attraction terms. Analysis of the glass atomic structure revealed a larger number of F1±O±F2 groups in the glass containing CaO (where Fi Al or B ions), although the total number of these groups remains lower than would be expected from a random ion distribution. The relatively higher Ca coordination number around Al compared with B ions shows a preferential Ca±Al approach compared with Ca±B. Finally, analysis of the Na and Ca environments showed that the Na ions more readily act as pure charge compensators ± i.e., distant from any non-bridging oxygen atoms ± than do the Ca ions, most of which conserve non-bridging oxygen atoms in their local environment. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 61.43Bn; 61.43Fs
1. Introduction The structure of multicomponent silicate oxide glasses is dicult to investigate because the techniques generally used for ordered structures are unsuitable for vitreous matter. Although the ®rst can be studied by co-ordination shell (1±2 A) various techniques (XAS, NMR, IR, Raman, etc.), is subject to the medium-range order (P5 A) considerable uncertainty, even if recent experi-
* Corresponding author. Tel.: +33-4 66 79 17 94; fax: +33-4 66 79 66 20. E-mail address: [email protected] (J.-M. Delaye).
ments [1] have suggested the presence of mediumscale organized structures. Simulation techniques provide an alternative to experimental methods for investigating the medium-scale order, notably molecular dynamics (MD), which is capable of reconstituting cells of a few cubic nanometers containing thousands (or millions) of atoms [2,3]. One of the essential aspects of simulation methodologies is the validity of the potentials used to describe atomic interactions: the potentials must reproduce the co-existence of covalent and ionic interactions in multicomponent glasses. This work adopted a novel approach coupling wide-angle X-ray diraction (WAXS) and MD
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 6 8 0 - 9
J.-M. Delaye et al. / Journal of Non-Crystalline Solids 293±295 (2001) 290±296
simulation with an intermediate reverse Monte Carlo (RMC) step [4] to ®t the interaction potentials on the experimental measurement data. The method has already been presented [5] without the potential ®tting step described here. An investigation of the relative interactions between cations (Na and Ca) and trivalent network formers (B and Al) is also proposed. 2. Methods 2.1. Molecular dynamics simulation and calculation of structure factors Interactions in vitreous oxide structures are classically represented in MD by Born±Mayer± Huggins potentials associating pair terms (Eq. (1) is the conventional formula to which the authors have added the 1=r6 and 1=r8 terms) and threebody angular terms (Eq. (2)) representing Coulomb interactions and covalent interactions [6,7]: ! rij qi qj C B U2 rij A exp ; 1 r6 r8 qij rij c c U3 rij ; rik ; hjik k exp rij rc rik rc cos hjik
2
cos h0 ;
2
where rij and rik are the interatomic distances, qi and qj the charges, and hjik the angle formed by an atom triplet j±i±k; the other parameters are adjustable parameters depending on the nature of the atoms involved. The Coulomb interactions are calculated using formal charges. The long-range component of these interactions is taken into account by the complete Ewald sum. An initial version of the Na± O and Ca±O potentials with zero B and C parameters was corrected by introducing 1=r6 and 1=r8 terms to obtain a better ®t of the depth of the potential well with the experimental data. The other pair potentials contain no 1=r6 or 1=r8 terms. The glass structures (Table 1) were prepared from a random con®guration. The liquid was stabilized between 4000 and 6000 K, before quenching at a rate of 1015 K s 1 to 1400 K, then
291
Table 1 Glass compositions (mol%) Glass 1 Glass 2
SiO2
B2 O3
Na2 O
ZrO2
Al2 O3
CaO
64.13 60.12
16.81 16.0
13.27 12.64
1.78 1.7
4.01 3.82
± 5.72
at 4 1014 K s 1 from 1400 K to room temperature. The atom positions were averaged over 3000 time steps to calculate the structure factor and for structure analysis. The structure factor was calculated from Eqs. (3) and (4): X S Q Wab QSab Q; 3 a;b
Wab Q;E
ca cb R fa Q;Efb Q;E jhf Q;Eij
2 dab ; 4
where S Q is the total structure factor, Sab Q the partial structure factors and Wab Q weighting factors depending on the atomic fraction (ci ) and on the atomic diraction factors (fi ) of species a and b. 2.2. Glass preparation and WAXS measurements The glasses in Table 1 were prepared by mixing reagent grade oxides. The powders were melted at 1100°C in graphite crucibles and quenched by partial immersion of the crucible. The glasses were then annealed for 1 h at 520 °C. Glass samples were cut as slabs 3 cm in diameter and 4 mm thick. WAXS measurements were performed on a diffractometer equipped with Mo Ka radiation and a bent graphite monochromator (Philips, PW 1729). The intensity measurements were carried out by the h=2h step scanning method with a constant step of 0.25° (2h) and in the angular range 0:5° < 2h < 140°, which corresponds to a Q range 1 (Q 4p sin h=k is the scatfrom 0.8 to 16:6 A tering factor, 2h is the scattering angle and k is the radiation wavelength). The total count accumulated at each measurement point was not less than 20 000. The X-ray source was operated at a current of 35 mA with an accelerating voltage of 50 kV. After correction (for polarization and absorption factors) and normalization, the coherently scattered intensity per atom was used to calculate the total structure factor S Q.
292
J.-M. Delaye et al. / Journal of Non-Crystalline Solids 293±295 (2001) 290±296
2.3. Potential ®tting method A two-step procedure was adopted in which the potentials were ®tted to the experimental results, then the glass structures were analyzed. Knowing the atomic positions from the MD model, the structure factors, S Q, for the simulation are calculated according to the Eqs. (3) and (4). The MD models are then used as the starting con®guration in a RMC modeling of the experimental structure factors. The partial pair distribution functions calculated from the MD and the RMC models are compared, which allow the determination of the interatomic potentials to modify. The RMC technique has already been used to ®t the atomic structure to experimental spectra [1]. It consists in minimizing the mean square deviation between the experimental and simulated structure factors by a series of random atom dis-
placements. The minimum approach distances between two types of atoms are imposed, and constraints are placed on Si and Al atoms to conserve the tetrahedral environments (the Al tetrahedral co-ordination in these glasses was recently demonstrated experimentally in [8]). There are no constraints for the other co-ordinations. The size of the cell is taken equal to that determined by the MD simulations. Fig. 1 compares the experimental structure factors with the values simulated by MD and after RMC correction. The deviations ± notably the initial double peak on the simulated structure factors ± disappear after the atomic displacements generated by the RMC technique. Fig. 2 shows the partial radial distribution functions for Na±O and Ca±O in glass 2 (the more complex one) as calculated by MD and re®ned by the RMC method. The structure corrected by RMC exhibits much better de®ned local orders
Fig. 1. Experimental structure factors compared with MD and RMC results.
Fig. 2. Partial Na±O and Ca±O radial distribution functions calculated by MD before potential ®tting, compared with the values for the RMC-corrected structure. The curves are shifted for convenience.
J.-M. Delaye et al. / Journal of Non-Crystalline Solids 293±295 (2001) 290±296
Fig. 3. Na±O and Ca±O potentials before and after ®tting (energy in 10
10
293
cgs).
Fig. 4. Comparison of experimental structure factor for glass 2 with ®tted MD values and with RMC values.
(higher intensity peaks) around the Na and Ca atoms, and the ®rst-neighbor distances are shorter than predicted by the initial MD model. We therefore attempted to deepen the Na±O and Ca± O potential wells by adding 1=r6 and 1=r8 terms and reducing the ®rst-neighbor distances. The curves for the Na±O and Ca±O potential pairs are plotted in Fig. 3 before and after ®tting the additional terms; with the deeper potential wells the simulated local orders are better de®ned around the Na and Ca atoms. Other changes included deleting the three-body term for the O±B±O triplets (necessary to better represent the boron coordination recently determined by 11 B NMR measurement) and slightly extending the Si±O ®rst-neighbor distance by increasing the Si±O repulsion. We recalculated the structure factors of the 6-oxide glass by MD using the readjusted potentials, and obtained a signi®cantly better agreement
with the experimental ®ndings (Fig. 4): the ®rst structure factor peak is no longer separated into two smaller peaks, and largely corresponds to the experimental peak. RMC was again used to re®ne the results: after RMC correction (Fig. 4) the simulated and experimental structure factors fully coincide. 3. Results The structures of 5- and 6-oxide glass compositions prepared by MD with ®tted potentials were analyzed, with particular attention to the interactions between the trivalent network formers (Al and B) incorporated in the polymerized network and cations (Na and Ca). The two compositions containing only Na2 O for the ®rst and a mixing Na2 O CaO for the second allow to compare the individual behaviors of Na and Ca. Table 2
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Table 2 Number of F1±O±F2 groups (where F Al or B) in glasses 1 and 2 (5184 atoms) Glass 1 Glass 2
Al±O±Al
Al±O±B
B±O±B
3
8 13
93 122
247 243
42 23
Table 3 Coordination numbers of network formers Al and B with Na radius and Ca within a 4.2 A Glass 1 Glass 2
Al(Na)
Al(Ca)
B(Na)
B(Ca)
1.94 1.9
± 0.57
2.43 2.46
± 0.41
indicates the number of Al±O±Al, B±O±B and Al± O±B groups in the glasses, considering only the oxygen atoms with two neighboring formers. Table 3 lists the Na and Ca co-ordination numbers around the Al and B atoms: Al(Na), for example, corresponds to the number of Na atoms within a spherical radius around the Al. In the fol4.2 A lowing text, the letter F will represent the formers Al or B, and M the modi®ers Na or Ca. The 4.2 A radius corresponds approximately to the minimum following the ®rst peak of the F±M partial radial distribution functions. Adding CaO to the 6-oxide glass had practically no eect on the F(Na) co-ordination number, although the Al(Ca)/Al(Na) ratio was nearly twice the B(Ca)/B(Na) ratio. This observation suggests that Ca tends to be situated near Al atoms rather than near B atoms. The number of F1±2 O±F2 groups that would be obtained in a random distribution can be estimated as follows (2 O signi®es that only oxygen atoms with two and only two neighboring formers are taken into account). Let NO2 represent the number of oxygen atoms with two neighboring formers (which also corresponds to the total number of groups as de®ned here), NF the total number of formers (including Si, B, Al and Zr) multiplied by their average co-ordinations, NF1
B±O±3 B
3
B±O±4 B
136 123
4
B±O±4 B
69 97
the number of type F1 formers multiplied by the F1 co-ordination, and NF2 the number of type F2 formers multiplied by the F2 co-ordination. If F1 and F2 are identical species, a random distribution of network formers would result in a number of F1±O±F2 triplets equal to NO2
NF1 NF1 1 : NF NF 1
5
If F1 and F2 are dierent species, this equation becomes NO2
NF1 NF2 2: NF NF 1
6
Table 4 indicates the number of groups corresponding to a random distribution. In the 5-oxide glass, the number of Al±2 O±Al and Al±2 O±B groups is well below the value expected from a random distribution of network formers. The exclusion principle known in the case of Al as the Lowenstein exclusion rule [9,10] is con®rmed in this glass, and is also applicable to Al±2 O±B triplets. The number of these groups increases with the addition of CaO, but remains below the result expected from a random distribution. Table 4 contains the number of 4 B±O±4 B, 3 B±O±3 B and 3 B±O±4 B groups in the simulated structures assuming a random distribution of 3 B and 4 B (representing 3- and 4-coordinate boron, respectively). The 4 B±O±4 B groups, under represented in the 5-oxide glass, are found slightly under the `normal' amount in the 6-oxide glass. Parallely the 3 B±O±4 B groups are favored in both compositions.
Table 4 Number of F1±O±F2 groups in glasses 1 and 2 assuming a random distribution of B and Al around oxygen atoms Glass 1 Glass 2
Al±O±Al
Al±O±B
B±O±B
3
B±O±3 B
15.7 17
125.3 128.2
248.8 241
34.5 26.5
3
B±O±4 B
116.4 107
4
B±O±4 B
97.9 107.6
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4. Discussion We have proposed a novel methodology for ®tting MD potentials to experimental ®ndings. The RMC technique directly reveals the pair potentials requiring correction. It is important to note that in the RMC results the experimental resolution is included. This explains partly the broadening of the peaks in the RMC ®ts compared to the initial MD model. The main dierence concerned the local environments of the Na and Ca cations. It was necessary not only to shorten the ®rst-neighbor distance, but also to order the shell of oxygen atoms around the cations by deepening the pair potential well. Using only Coulomb attraction and exponential repulsion terms alone was not sucient to obtain a satisfactory ®t, hence the additional 1=r6 and 1=r8 terms. This work involves only the atomic structure of the glasses, which we ®t to the experimental structure factors. We did not verify the quality of the potentials for representing elastic or dynamic quantities. The satisfactory reproduction of the experimental structure means that the relative intensities of the atomic interactions are correct, although some deviations may remain with respect to the actual intensities. In particular, hardening the Na±O and Ca±O interactions allowed us to limit the deviation with the intensity of the formeroxygen interactions. The need for this correction probably arises from the use of formal charges. The Coulomb component of the former-oxygen interactions is overestimated because the interactions are considered as purely ionic, whereas the experimental results and electron structure calculations indicate lower estimated charge values [11±13]: for example, ab initio calculations for Si and O clusters [14] yielded charges on the order of 1.6 for Si and about )0.8 for O. An ionic representation with formal charges in addition to the angular term intended to represent the rigidity of the covalent bonds, thus overestimates the former-oxygen interactions compared with the modi®er-oxygen interactions. The corrections implemented to deepen the Na±O and Ca±O potential wells thus compensate for the overestimated former-oxygen interactions.
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Separating the repulsive eect into an exponential term and 1=r8 term tends to harden the short-range repulsion, while the 1=r6 term hardens the attractive eect. The combination of the two eects creates a deeper potential well. For the moment we consider these extra terms to be only a mathematical tool for re®ning the potentials, with no special signi®cance attached to the multipolar interactions the terms might contain. Fitting the new attraction terms qM with a pure Coulomb qO =r C=r6 from 2 to 3 A term (qeff qO =r) nearly doubles the eective charges qeff on the Na and Ca cations, resulting in an order-of-magnitude agreement between the formal Si charge (+4) and the actual charge of about +1.6. The relative interactions between Al, B, Ca and Na were elucidated by analysis of the atomic structures. The Ca ions tend to approach Al, rather than B atoms. The interaction potentials used do not take into account chemical interactions between elements. Space limitations may account for the preference of Ca ions for the vicinity of Al atoms: the AlO4 groups are larger than the BO4 groups (the ®rst-neighbor distance Al±O is around while the B±O distance is around 1.4 A). 1.75 A Adding CaO also increases the number of Al±2 O±Al, Al±2 O±B and 4 B±2 O±4 B groups (see the de®nition above). The Lowenstein exclusion law appears less applicable in the glass containing CaO than in the 5-oxide glass, although the number of groups remains below the values predicted for a random distribution. These results are consistent with ab initio calculations that show the easier formation of Al±O±Al groups when Ca is present [15]. The double charge of the Ca ions thus favors a closer approach to the trivalent formers in tetrahedral coordination. However, the number of B±O±B groups in both glasses is practically equal to the value expected for a random distribution. This result shows no segregation of boron but is not consistent with recent NMR analyses showing 3 B mixed with the silicate network [16]. The very fast quenching rates applied by MD are poorly suited to a study long-range diusion processes than can be responsible for this phenomenon. Finally, consider the role of Na and Ca as charge compensators or modi®ers. We designated
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as charge compensators all the Na or Ca ions with rano non-bridging oxygen atoms within a 3.4 A dius. In the 5-oxide glass, 57% of the Na ions were charge compensators, compared with 47% of the Na ions and 16% of the Ca ions in the 6-oxide glass. The Ca ions were always closer to the nonbridging oxygens than the Na ions, as con®rmed by the non-bridging oxygen co-ordination numbers around the Na (0.72) and Ca (1.75) in the 6oxide glass. It seems rather paradoxical that adding CaO favors the formation of F1±2 O±F2 groups, whereas the percentage of charge-compensating Ca ions with no nearby non-bridging oxygen atoms remains low. The explanation probably lies in the double charge of the Ca ions, with more ambiguous eects than the single-charged Na ions: the Ca ions can approach non-bridging oxygen atoms without losing their charge-compensating role. Their greater volume probably prevents them from being incorporated in large quantities in the polymerized network as pure charge compensators.
5. Conclusion An innovative method coupling MD and RMC analysis allowed us to ®t the interaction potentials on the experimental structure factors of aluminoborosilicate glass compositions obtained by WAXS. By calculating the structure factors and the partial radial distribution functions, the combination of RMC and MD directly identi®es the pair potentials requiring optimization. We observed that a Coulomb potential alone is not enough to model a suciently ordered local environment around the Na and Ca cations: additional terms were necessary to ®t the repulsion and attraction terms. Analysis of the glass atomic structure revealed that with only Na2 O as modi®ers in the composition, the Loewenstein rule is veri®ed not only for Al±O±Al groups but also for Al±O±B and
4
B±O±4 B groups. With Na2 O CaO, a larger number of Al±O±Al, Al±O±B and 4 B±O±4 B groups are calculated, although the total number of these groups remains lower than would be expected from a random ion distribution. The relatively higher Ca coordination number around Al compared with B ions shows a preferential Ca±Al approach compared with Ca±B. Finally, analysis of the Na and Ca environments showed that Na ions more readily act as pure charge compensators ± i.e., distant from any non-bridging oxygen atoms ± than do the Ca ions, most of which conserve non-bridging oxygen atoms in their local environment. Their role as network modi®ers and charge compensators is less clearly de®ned than for the Na ions. References [1] L. Cormier, P.H. Gaskell, G. Calas, A.K. Soper, Phys. Rev. B 58 (1998) 11322. [2] J.-M. Delaye, D. Ghaleb, J. Non-Cryst. Solids 195 (1996) 239. [3] J.-M. Delaye, V. Louis-Achille, D. Ghaleb, J. Non-Cryst. Solids 210 (1997) 232. [4] R.L. McGreevy, Nucl. Instrum. and Meth. A 354 (1995) 1. [5] L. Cormier, D. Ghaleb, J.-M. Delaye, G. Calas, Phys. Rev. B 61 (2000) 14495. [6] T.F. Soules, J. Non-Cryst. Solids 49 (1982) 29. [7] F.H. Stillinger, T.A. Weber, Phys. Rev. B 43 (1991) 1194. [8] B. Boizot, G. Petite, D. Ghaleb, N. Pellerin, F. Fayon, B. Reynard, G. Calas, Nucl. Instrum. and Meth. B 166&167 (2000) 502. [9] S. Wang, J.F. Stebbins, J. Am. Ceram. Soc. 82 (1999) 1519. [10] W. Loewenstein, Am. Mineral. 39 (1954) 92. [11] C.H. Hsieh, H. Jain, A.C. Miller, E.I. Kamitsos, J. NonCryst. Solids 168 (1994) 247. [12] A.C. Lasaga, G.V. Gibbs, Phys. Chem. Mineral. 14 (1987) 107. [13] K. De Boer, A.P.J. Jansen, R.A. Van Santen, Chem. Phys. Lett. 223 (1994) 46. [14] V.S. Yushchenko, E.D. Shchukin, M. Hotokka, J. Mater. Sci. 29 (1994) 3038. [15] J.A. Tossell, G. Saghi-Szabo, Geochim. Cosmochim. Acta 61 (1997) 1171. [16] S. Wang, J.F. Stebbins, J. Non-Cryst. Solids 231 (1998) 286.