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Computer-modelling studies of 4-2 coordinated glasses
22
Journal of Non-Crystalline Solids 123 (1990) 22-25 North-Holland
COMPUTER-MODELLING STUDIES OF 4-2 COORDINATED GLASSES L.F. G L A D D E N Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge, CB2 3RA, UK
Completely computer-generated models of 4-2 coordinated glasses have been constructed using an algorithm which allows a systematic investigation of the parameter-space describing amorphous networks. This paper describes the application of this algorithm to two very different glasses in the AX 2 system: SiSe2 which is found to be based on a structure consisting of randomly oriented chains of edge-sharing Si(Se1,2)4 tetrahedra and six-membered rings, and SiO2whose structure is consistent with a continuous random network of comer-sharing tetrahedra. It is now possible to modify these algorithms for the study of GeS2 and GeSe2 and hence to investigate the possible existence of 'outrigger rafts' and rings in these glasses.
1. In~oduction The major objectives in designing computer algorithms to generate models of amorphous networks are: (i) to maintain control over the parameters which could affect the model so that a complete investigation of parameter space and of the topology of the final structure is possible; (ii) to achieve a high degree of connectivity without introducing undue strains into the structure and to build large models in which edge-effects are minimized. Models contain - 1 0 0 0 atoms; (iii) the need to include specific structural features in the model such as rings and regions of crystallinity. The approach which has been adopted is to simulate on the computer the procedure which an experimenter would adopt in building a ball-andstick model, a method which has proved successful in constructing small networks. The computerbased method has all the advantages of the balland-stick approach (high degree of connectivity, rigid structural units which automatically reproduce short-range order) but, in addition, has a number of significant advantages: (a) 'Unbiasedness' during model construction. The model-building proceeds to a well-defined set of rules. (b) Large models can be generated quickly,
allowing a thorough investigation of parameterspace by producing a range of models built with different structural parameters and structural units. (c) The coordinates of the models are already in numerical form. Subsequent stages of the analysis such as energy relaxation and the determination of radial distribution functions is therefore quick and routine. Central to the algorithms is the concept of adding a rigid structural unit (AnXm) to the existing structure thereby maintaining stoichiometry and short-range order. A second aspect of the algorithms is their ability to 'unwind' earlier additions to the structure if, at a later stage of the model construction, such units are found to impede further development of the network. After construction all the models are energy relaxed using the Keating expression for local strain energy, with an additional L e n n a r d - J o n e s 12-6 potential approximating non-bonded interactions. The two models described in this paper differ significantly at a structural level - SiO 2 is a C R N of coruer-sharing tetrahedral units, while SiSe 2 is a chain-like structure of edge-sharing units. Therefore, two algorithms have been developed, one based on each of these geometries. Although they are applied here to SiO 2 and SiSe 2, respectively, the approach is quite general and is easily extended to other glass-forming systems.
0022-3093/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
L.F. Gladden / Computer-modelling studies o/4-2 coordinated glasses
23
2. Results and discussion
2.1. Corner-sharing tetrahedral networks (Si02) The construction of the corner-sharing C R N proceeds by the addition of AX 3 units to the existing structure. As with the edge-sharing tetrahedra algorithm there are no unsatisfied ' A ' - a t o m bonds during building, and any unsatisfied ' X ' atom bond is regarded as a possible site for the addition of the new unit. Figure 1 shows the geometric constraints relevant to the addition of an AX 3 unit. There are two degrees of freedom namely rotations about the A ° - X ° bond and the X ° - A ' bond which together allow the AX 3 unit to explore a large number of possible orientations in space. To maintain a fully connected C R N (i.e. fully valence-satisfied structure) it must be possible for the new AX 3 unit to join to the existing network via two or more A - X bonds. A number of positions for the A' atom are tested (typically thirty 20 ° rotations about A ° - X °) and for each, rotations about the X ° - A ' b o n d give possible sites for the X ' atoms (typically ten 20 ° rotations - note the 120 ° rotational symmetry about X ° - A ' ) . For each of these 300 locations the following tests are made. (a) Are the A' and all X ' atoms further than their exclusion spheres from the existing atoms in the network? If the answer is 'yes', there is no possible join. (b) Is there an overlap between an X ' a t o m and an existing X atom (X B) with an unsatisfied
,o..
'
Fig. 1. The basic geometry used in the addition of an AX 3 unit (A' and X' atoms) during construction of a CRN network. The degrees of freedom about the A°-X ° and X°-A ' bonds are indicated. X B is a possible atom to which a 'join' could be made.
20
'2
'4
46
r (/~) Fig. 2. T(r) for the final SiO2 model ( experiment (+) [1].
8
10
compared with
bond? If the answer is 'yes', a join is possible. A calculation is then performed to determine how strained the join would be and is quantified in terms of the deviations of the X ° - A ' and A ' - X B bond lengths and the X ° - A ' - X B, X ' - X R-AB bond angles from the respective equilibrium values describing the glass. Further, the position of the two X ' atoms not involved in the initial join are tested to ensure either that they are sufficiently distant from the existing C R N or that they are able to make further joins. Each of (a) and (b) are also considered with neighbouring AX 3 units (which are only joined by a single A - X bond) removed from the model - this is an example of 'unwinding'. Having completed the investigation for all possible A', X ' combinations a decision is made as to which location is ' b e s t ' - for SiO z the decision is made subject to the following conditions: (i) Accept a join if the percentage deviation from an 'ideal' join is less than a pre-defined value. Then for all accepted joins choose that join which has the smallest deviation and the largest non-bonded X ' to existing C R N distances. (ii) If we do not have an acceptable join choose that location providing the largest non-bonded A' or X ' to existing C R N distances. (iii) If both (i) and (ii) fail it is not possible to develop the C R N from this site - we are left with a 'dangling' bond. After a join is made a few ( - 5) iterations of the relaxation algorithm are performed to remove residual strains. The next site for the addition of an AX 3 unit is chosen to be that nearest the centre of the model. U p o n completion of the model the final structure is energy-relaxed. Figure 2 shows the T(r) calculated from a pure C R N model in comparison with the data of Wright and Sinclair [1]. Figure 3 shows the S i - O - S i bond-angle distil-
LF. Gladden / Computer-modellingstudies of 4-2 coordinatedglasses
24 1.0 b 0.8
-~ 0.6
~ o.4 ~ 0.2 0.0
120
140
160 180 ansle Fig. 3. The histogram shows the Si-O-Si bond-angle distribution for the final model whose T(r) is shown in fig. 2. The bond-angle distribution obtained from 29Si MAS-NMR [5] ( ) and selected points from the X-ray data of Mozzi and Warren [2] (It) are shown for comparison. Si-O-Si
bond
bution for this network. The bond-angle distribution is in good agreement with that obtained from the X-ray study of Mozzi and Warren [2], indicating a peak in the bond-angle distribution at - 143 °. Further, this result is entirely consistent [3,4] with the 29Si chemical shift of vitreous SiO2 which occurs at - 1 1 1 . 5 ppm, referenced to tetramethylsilane [5].
2.2. Edge-sharing tetrahedral networks (SiSe 2) The edge-sharing tetrahedral network is the easier of the two algorithms to develop. The only flexibility allowed is at the 'hinge' formed by the c o m m o n edge of two rigid AX 4 units (see fig. 4) which provides only one degree of freedom. It is
Y2
X'
therefore essential to be able to ' u n w i n d ' the chain back at least one unit during the construction to the previous 'hinge' (which is at right angles to the one under consideration) to provide sufficient flexibility in the chain development. The ability to remove short chains, which m a y impede the growth of longer ones, is also included in the model. Construction proceeds from 'seeds', which may represent specific structural features, such as rings, dispersed at r a n d o m in the modelling volume from which chains are grown until the required density is reached [6]. The parameters which m a y be altered within the model construction are: (1) size of the model (set by computing restrictions); (2) n u m b e r of starting chains; (3) n u m b e r and size of 'seeds' and the combination of 'seed' types; (4) geometrical variables of the basic unit determined by the need to match low r in the T ( r ) ; (5) m a x i m u m variation in the bending angle between edge-sharing tetrahedra; (6) m i n i m u m chain length; (7) dihedral angle fluctuation, (8) exclusion spheres around atoms (defined by covalent radii); (9) final density of the model. R a n d o m chains longer than 10 tetrahedral units ( - 30 A) are c o m m o n with some as long as 15-20 units. Most importantly, nearly all chains are longer than the diameter of the 20 ,~ sphere from which the T(r) and Qi(Q) distributions are
T(r) 2
0 6
b
.
.
.
i
. . . .
i
. . . .
i
,
•
J
. . . .
T(r) 2 0 x1
I
x2 Fig. 4. The geometry of adding an edge-sharing AX4 unit to an existing chain. The single degree of freedom is indicated as a rotation about the x 1- Yl' hinge'. To provide sufficient freedom for building, the chain is 'unwound' to the x2- y2'hinge'.
....
12 . . . .
14. . . . .
16 . . . .
18 . . . .
10
(-&) Fig. 5. Comparison of T(r) calculated for (a) a chain-based model containing -15% of its atoms in six-membered rings and (b) a CRN model, with the nentron-scattering data of Johnson [7] ( + ).
L F. Gladden / Computer-modellingstudies of 4-2 coordinated glasses
calculated thereby suggesting a realistic model of a-SiSe2 has been produced with very few chain ends in the structure. Figure 5 compares the best model built using the above procedure with a comer-sharing CRN model produced by taking the coordinates of the Gaskell and Tarrant model and replacing the oxygen atoms with selenium atoms. The Si-Se-Si bond angle and Si-Se bond length were then varied, assuming the Se-Si-Se angle to be the pure tetrahedral angle. It is clear that in the CRN model the feature occurring at - 2 . 8 /k and the third peak in the experimental T ( r ) are both missing or, at best, extremely ill-defined. A CRN description of a-SiSe2 is found to be inappropriate in this study. The model giving best agreement with experimental data contains - 1 5 % of the atoms in the total model in six-membered tings, thereby introducing some corner-sharing tetrahedra into the glass structure. Good agreement with experiment in the 6-8 A, region also requires a degree of alignment between the chains of edge-
25
sharing tetrahedra. A detailed discussion of this modelling study has been reported elsewhere [810l.
References [1] A.C. Wright and R.N. Sinclair, J. Non-Cryst. Solids 76 (1985) 351. [2] R.L. Mozzi and B.E. Warren, J. Appl. Crystallogr. 2 (1969) 164. [3] C. Fiori and R.A.B. Devine, Mater. Res. Soc. Symp. Proc. 61 (1986) 187. [4] J.V. Smith and C.S. Blackwell, Nature 303 (1983) 223. [5] L.F. Gladden, T.A. Carpenter and S.R. Elliott, Philos. Mag. B53 (1986) L81, [6] R.W. Johnson S. Susman J. McMillan and K.J. Volin, Mater. Res. Bull. 21 (1986) 41. [7] R.W. Johnson, J. Non-Cryst. Solids 88 (1986) 366. [8] L.F. Gladden and S.R. Elllott, Phys. Rev. Lett. 59 (1987) 908. [9] L.F. Gladden and S.R. EUiott, J. Non-Cryst. Solids 109 (1989) 211. [10] L.F. Gladden and S.R. Elliott, J. Non-Cryst. Solids 109 (1989) 223.
Journal of Non-Crystalline Solids 123 (1990) 22-25 North-Holland
COMPUTER-MODELLING STUDIES OF 4-2 COORDINATED GLASSES L.F. G L A D D E N Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge, CB2 3RA, UK
Completely computer-generated models of 4-2 coordinated glasses have been constructed using an algorithm which allows a systematic investigation of the parameter-space describing amorphous networks. This paper describes the application of this algorithm to two very different glasses in the AX 2 system: SiSe2 which is found to be based on a structure consisting of randomly oriented chains of edge-sharing Si(Se1,2)4 tetrahedra and six-membered rings, and SiO2whose structure is consistent with a continuous random network of comer-sharing tetrahedra. It is now possible to modify these algorithms for the study of GeS2 and GeSe2 and hence to investigate the possible existence of 'outrigger rafts' and rings in these glasses.
1. In~oduction The major objectives in designing computer algorithms to generate models of amorphous networks are: (i) to maintain control over the parameters which could affect the model so that a complete investigation of parameter space and of the topology of the final structure is possible; (ii) to achieve a high degree of connectivity without introducing undue strains into the structure and to build large models in which edge-effects are minimized. Models contain - 1 0 0 0 atoms; (iii) the need to include specific structural features in the model such as rings and regions of crystallinity. The approach which has been adopted is to simulate on the computer the procedure which an experimenter would adopt in building a ball-andstick model, a method which has proved successful in constructing small networks. The computerbased method has all the advantages of the balland-stick approach (high degree of connectivity, rigid structural units which automatically reproduce short-range order) but, in addition, has a number of significant advantages: (a) 'Unbiasedness' during model construction. The model-building proceeds to a well-defined set of rules. (b) Large models can be generated quickly,
allowing a thorough investigation of parameterspace by producing a range of models built with different structural parameters and structural units. (c) The coordinates of the models are already in numerical form. Subsequent stages of the analysis such as energy relaxation and the determination of radial distribution functions is therefore quick and routine. Central to the algorithms is the concept of adding a rigid structural unit (AnXm) to the existing structure thereby maintaining stoichiometry and short-range order. A second aspect of the algorithms is their ability to 'unwind' earlier additions to the structure if, at a later stage of the model construction, such units are found to impede further development of the network. After construction all the models are energy relaxed using the Keating expression for local strain energy, with an additional L e n n a r d - J o n e s 12-6 potential approximating non-bonded interactions. The two models described in this paper differ significantly at a structural level - SiO 2 is a C R N of coruer-sharing tetrahedral units, while SiSe 2 is a chain-like structure of edge-sharing units. Therefore, two algorithms have been developed, one based on each of these geometries. Although they are applied here to SiO 2 and SiSe 2, respectively, the approach is quite general and is easily extended to other glass-forming systems.
0022-3093/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
L.F. Gladden / Computer-modelling studies o/4-2 coordinated glasses
23
2. Results and discussion
2.1. Corner-sharing tetrahedral networks (Si02) The construction of the corner-sharing C R N proceeds by the addition of AX 3 units to the existing structure. As with the edge-sharing tetrahedra algorithm there are no unsatisfied ' A ' - a t o m bonds during building, and any unsatisfied ' X ' atom bond is regarded as a possible site for the addition of the new unit. Figure 1 shows the geometric constraints relevant to the addition of an AX 3 unit. There are two degrees of freedom namely rotations about the A ° - X ° bond and the X ° - A ' bond which together allow the AX 3 unit to explore a large number of possible orientations in space. To maintain a fully connected C R N (i.e. fully valence-satisfied structure) it must be possible for the new AX 3 unit to join to the existing network via two or more A - X bonds. A number of positions for the A' atom are tested (typically thirty 20 ° rotations about A ° - X °) and for each, rotations about the X ° - A ' b o n d give possible sites for the X ' atoms (typically ten 20 ° rotations - note the 120 ° rotational symmetry about X ° - A ' ) . For each of these 300 locations the following tests are made. (a) Are the A' and all X ' atoms further than their exclusion spheres from the existing atoms in the network? If the answer is 'yes', there is no possible join. (b) Is there an overlap between an X ' a t o m and an existing X atom (X B) with an unsatisfied
,o..
'
Fig. 1. The basic geometry used in the addition of an AX 3 unit (A' and X' atoms) during construction of a CRN network. The degrees of freedom about the A°-X ° and X°-A ' bonds are indicated. X B is a possible atom to which a 'join' could be made.
20
'2
'4
46
r (/~) Fig. 2. T(r) for the final SiO2 model ( experiment (+) [1].
8
10
compared with
bond? If the answer is 'yes', a join is possible. A calculation is then performed to determine how strained the join would be and is quantified in terms of the deviations of the X ° - A ' and A ' - X B bond lengths and the X ° - A ' - X B, X ' - X R-AB bond angles from the respective equilibrium values describing the glass. Further, the position of the two X ' atoms not involved in the initial join are tested to ensure either that they are sufficiently distant from the existing C R N or that they are able to make further joins. Each of (a) and (b) are also considered with neighbouring AX 3 units (which are only joined by a single A - X bond) removed from the model - this is an example of 'unwinding'. Having completed the investigation for all possible A', X ' combinations a decision is made as to which location is ' b e s t ' - for SiO z the decision is made subject to the following conditions: (i) Accept a join if the percentage deviation from an 'ideal' join is less than a pre-defined value. Then for all accepted joins choose that join which has the smallest deviation and the largest non-bonded X ' to existing C R N distances. (ii) If we do not have an acceptable join choose that location providing the largest non-bonded A' or X ' to existing C R N distances. (iii) If both (i) and (ii) fail it is not possible to develop the C R N from this site - we are left with a 'dangling' bond. After a join is made a few ( - 5) iterations of the relaxation algorithm are performed to remove residual strains. The next site for the addition of an AX 3 unit is chosen to be that nearest the centre of the model. U p o n completion of the model the final structure is energy-relaxed. Figure 2 shows the T(r) calculated from a pure C R N model in comparison with the data of Wright and Sinclair [1]. Figure 3 shows the S i - O - S i bond-angle distil-
LF. Gladden / Computer-modellingstudies of 4-2 coordinatedglasses
24 1.0 b 0.8
-~ 0.6
~ o.4 ~ 0.2 0.0
120
140
160 180 ansle Fig. 3. The histogram shows the Si-O-Si bond-angle distribution for the final model whose T(r) is shown in fig. 2. The bond-angle distribution obtained from 29Si MAS-NMR [5] ( ) and selected points from the X-ray data of Mozzi and Warren [2] (It) are shown for comparison. Si-O-Si
bond
bution for this network. The bond-angle distribution is in good agreement with that obtained from the X-ray study of Mozzi and Warren [2], indicating a peak in the bond-angle distribution at - 143 °. Further, this result is entirely consistent [3,4] with the 29Si chemical shift of vitreous SiO2 which occurs at - 1 1 1 . 5 ppm, referenced to tetramethylsilane [5].
2.2. Edge-sharing tetrahedral networks (SiSe 2) The edge-sharing tetrahedral network is the easier of the two algorithms to develop. The only flexibility allowed is at the 'hinge' formed by the c o m m o n edge of two rigid AX 4 units (see fig. 4) which provides only one degree of freedom. It is
Y2
X'
therefore essential to be able to ' u n w i n d ' the chain back at least one unit during the construction to the previous 'hinge' (which is at right angles to the one under consideration) to provide sufficient flexibility in the chain development. The ability to remove short chains, which m a y impede the growth of longer ones, is also included in the model. Construction proceeds from 'seeds', which may represent specific structural features, such as rings, dispersed at r a n d o m in the modelling volume from which chains are grown until the required density is reached [6]. The parameters which m a y be altered within the model construction are: (1) size of the model (set by computing restrictions); (2) n u m b e r of starting chains; (3) n u m b e r and size of 'seeds' and the combination of 'seed' types; (4) geometrical variables of the basic unit determined by the need to match low r in the T ( r ) ; (5) m a x i m u m variation in the bending angle between edge-sharing tetrahedra; (6) m i n i m u m chain length; (7) dihedral angle fluctuation, (8) exclusion spheres around atoms (defined by covalent radii); (9) final density of the model. R a n d o m chains longer than 10 tetrahedral units ( - 30 A) are c o m m o n with some as long as 15-20 units. Most importantly, nearly all chains are longer than the diameter of the 20 ,~ sphere from which the T(r) and Qi(Q) distributions are
T(r) 2
0 6
b
.
.
.
i
. . . .
i
. . . .
i
,
•
J
. . . .
T(r) 2 0 x1
I
x2 Fig. 4. The geometry of adding an edge-sharing AX4 unit to an existing chain. The single degree of freedom is indicated as a rotation about the x 1- Yl' hinge'. To provide sufficient freedom for building, the chain is 'unwound' to the x2- y2'hinge'.
....
12 . . . .
14. . . . .
16 . . . .
18 . . . .
10
(-&) Fig. 5. Comparison of T(r) calculated for (a) a chain-based model containing -15% of its atoms in six-membered rings and (b) a CRN model, with the nentron-scattering data of Johnson [7] ( + ).
L F. Gladden / Computer-modellingstudies of 4-2 coordinated glasses
calculated thereby suggesting a realistic model of a-SiSe2 has been produced with very few chain ends in the structure. Figure 5 compares the best model built using the above procedure with a comer-sharing CRN model produced by taking the coordinates of the Gaskell and Tarrant model and replacing the oxygen atoms with selenium atoms. The Si-Se-Si bond angle and Si-Se bond length were then varied, assuming the Se-Si-Se angle to be the pure tetrahedral angle. It is clear that in the CRN model the feature occurring at - 2 . 8 /k and the third peak in the experimental T ( r ) are both missing or, at best, extremely ill-defined. A CRN description of a-SiSe2 is found to be inappropriate in this study. The model giving best agreement with experimental data contains - 1 5 % of the atoms in the total model in six-membered tings, thereby introducing some corner-sharing tetrahedra into the glass structure. Good agreement with experiment in the 6-8 A, region also requires a degree of alignment between the chains of edge-
25
sharing tetrahedra. A detailed discussion of this modelling study has been reported elsewhere [810l.
References [1] A.C. Wright and R.N. Sinclair, J. Non-Cryst. Solids 76 (1985) 351. [2] R.L. Mozzi and B.E. Warren, J. Appl. Crystallogr. 2 (1969) 164. [3] C. Fiori and R.A.B. Devine, Mater. Res. Soc. Symp. Proc. 61 (1986) 187. [4] J.V. Smith and C.S. Blackwell, Nature 303 (1983) 223. [5] L.F. Gladden, T.A. Carpenter and S.R. Elliott, Philos. Mag. B53 (1986) L81, [6] R.W. Johnson S. Susman J. McMillan and K.J. Volin, Mater. Res. Bull. 21 (1986) 41. [7] R.W. Johnson, J. Non-Cryst. Solids 88 (1986) 366. [8] L.F. Gladden and S.R. Elllott, Phys. Rev. Lett. 59 (1987) 908. [9] L.F. Gladden and S.R. EUiott, J. Non-Cryst. Solids 109 (1989) 211. [10] L.F. Gladden and S.R. Elliott, J. Non-Cryst. Solids 109 (1989) 223.