J. CHEM. SOC. FARADAY TRANS., 1992, 88(8), 1161-1164 1161

Molecular Mechanics Study on the a-Quartzlp-Quartz Transition

Erik de Vos Burchart, Herman van Bekkum and Bastiaan van de Graaf Laboratory for Organic Chemistry and Catalysis , Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands

Molecular mechanics calculations have been performed on the alp-quartz phase transition. By applying a volume constraint the size of t he unit cell of a-quartz was changed. At a transition volume V,, = 0.1208 nm3 the symmetry of /?-quartz was achieved. Calculated volume-pressure data are in agreement with experimental data. The negative pressure applied at t h e transition volume (-3.35 GPa) is in agreement with the experimental relation between pressure and transition temperature (dTtr/dp). During the calculations t h e vibrational modes, t h e heat of formation and the structural parameters were followed a s a function of the constrained volume. From a molecular mechanics point of view p-quartz is shown to be the stable structure at unit-cell volumes larger than "tr .

The major forms of crystalline silica are quartz, tridymite and cristobalite with transition temperatures of 1140 and 1743 K. Within each form non-reconstructive transitions occur ; thus a-quartz transforms at 846 K into /?-quartz. In this reversible transition the density p decreases from 2.65 to 2.53 g C I I I - ~ , the Si-0-Si angles increase from 144" to 155", and the symmetry passes from threefold to sixfold according to the enantiomorphic pairs of space groups 04 or DZ (a-quartz, Brazil twinning) and Dg or Dz (/?-quartz).'V2

Despite much experimental work there is still some contro- versy on the classification of the a//?-quartz transition at 846 K. First, the structural classification of the transition could be either order-disorder or displacive. In the case of a dis- placive transition /?-quartz is a truly hexagonal structure. The order-disorder transition involves the two Dauphine twin structures (secondary twinning) of a-quartz with atoms hopping from one phase to the other, with the idealized p- quartz positions as time-averaged positions. Secondly, the thermodynamic classification could be first or second order. The observation of a transition enthalpy and a small hyster- esis and the presence of vibrational modes with decreasing frequency in the vicinity of the transition temperature (soft modes) supports the classification of the alp-quartz transition (neglecting the incommensurate phase) as a displacive, first- order transitions3

The quartz structure and its phase transition have recently been the subject of computer simulations. The geometry of the quartz structure has been studied by Tsuneyuki et ~ l . , ~ who concluded after molecular dynamics studies that the p- quartz structure is built of clusters of the two Dauphine twin structures. Atoms hop from one position to the other with competing factors being the thermal energy and the potential barrier separating the two a-quartz structures. Because of the sensitivity of the potential surface to the volume these authors have taken the thermal expansion into account.

Silvi et d5 came to the same conclusion using periodic pseudopotential Hartree-Fock calculations. These authors simulated the thermal expansion by a negative pressure ( - 1.5 GPa), which caused the barrier height to decrease, but the two minima representing the a-quartz structures remained. Vibrational aspects of the a//? transition have been studied by several workers6-' using valence force fields. The frequency of the vibration with wavenumber 207 cm- ' (room temperature) was calculated to vanish for the /?-quartz struc- turq6 although experimentally this has not been confirmed. '' Etchepare et 121.' added a torsional force to the valence force field, resulting in a decreasing, but non-vanishing, 207 cm- ' mode for the /?-quartz structure. The calculated eigenvector

of the 207 cm- vibration, containing the atomic displace- ments, is very similar to the transition vector, which describes the a//?-quartz

In the present work the structure of a-quartz is converted into the structure of /?-quartz by applying a volume con- straint on the structure using our molecular mechanics force field.12 This all-silica force field was recently developed and was shown to be consistent in calculating the heat of forma- tion, the vibrational frequencies (including IR intensities) and the geometry for structures ranging from the dense a-quartz structure to the open structure of all-silica faujasite. The force field is based on the high degree of covalence present in silica structures. The low ionicity is expressed by low partial charges (ca. 10% of the formal charges) and the relative per- mittivity used ( E = 6.2). The force field consists of the Morse potential to describe the bonds, the Urey-Bradley potential to describe the valency angles, and the Hill and Coulomb potentials to describe the long-range contributions to the total energy. The last interactions are calculated only to a distance of 1.4 nm using a taper function.

Recently our force field was used in combination with the MM3 force field for hydrocarbon^'^ to describe the struc- tural changes in the deformation of the zeolite MFI frame- work upon the adsorption of p-xylene. Both direction and size of the overall displacement vector of the framework atoms were predicted a~curate1y.l~

Computational Details Calculations were carried out on a DEC 5000/200 work- station using the Delft computer program for molecular mechanics. The full-matrix Newton-Raphson method16 was used in the energy minimization.

In this method the displacement vector dx for both carte- sian coordinates of all atoms and the lattice parameters is obtained in each iteration from the Hessian matrix H a n d the gradient V E by:

H * d x = - V E (1)

The Hessian matrix is the second-derivative matrix of the energy with respect to the coordinates of all atoms in the unit cell. Threefold singularity (translational degrees of freedom) is overcome by adding three Eckart constraints to keep the centre of mass fixed. The second derivatives of the energy with respect to the lattice parameters and mixed derivatives are also added to the set of linear equations in eqn. (1).

Energy minima are characterized by a zero gradient and a Hessian matrix, which has three zero eigenvalues

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1162 J. CHEM. SOC. FARADAY TRANS., 1992, VOL. 88

(translations) and (3N - 3) non-negative eigenvalues. One negative eigenvalue indicates a saddle point on the potential- energy surface. The optical vibrational modes are calculated from the mass-weighted Hessian matrix using a wavevector

The main differences in the structural parameters of the a and P quartz structures are the larger unit-cell volume and the higher symmetry (silicon positional parameter u = 0.5) of the P-quartz structure. As the unit-cell volume increases with temperature owing to thermal expansion, it is not unlikely that the transition is related to the increasing unit-cell volume.

The sensitivity of the potential surface of the quartz struc- ture to the unit-cell volume is assumed to be as is schemati- cally shown in Fig. 1. The two minima represent the Dauphine twin structures of a-quartz. With increasing volume the barrier separating the two minima is lowered and finally the two minima coalesce to form a single minimum, the P-quartz structure.

Constraints are an effective tool for modifying a structure in molecular mechanics calculations. We can distinguish two types of constraints. (i) Internal constraints : These can be used to move across the potential-energy surface over energy barriers to other minima. (ii) External constraints: These con- straints modify the potential-energy surface. Using an exter- nal constraint the energy minimum is never left: the position of the minimum is changed due to the sensitivity of the potential-energy surface to the constraint applied.

An example of an internal constraint is the forced rotation about the C-C bond in ethane. During the rotation the interatomic forces are out of equilibrium. An example of an external constraint is the application of pressure: the struc- ture will change so as to meet this external force. However, the forces between the atoms are still in equilibrium.

For the additional volume constraint, an external con- straint, an extra equation [eqn. (2)] was added to the set of linear equations in eqn. (1) :

k = 0.

3

Kctuat - Viesired = A‘ = C ( t i i t j j AtkJ (2)

where taB are elements of the transformation matrix (a, P = 1, 3).

The transformation matrix is the matrix which converts fractional coordinates to Cartesian coordinates. The product of the diagonal elements of this matrix is the actual volume during the calculations. The constraint is achieved when

i f j f k

A V = O .

/ a a \

Results and Discussion The structure of a-quartz obtained after energy minimization with our force field, without any constraint applied, is found to be a true energy minimum with respect to the gradient and the eigenvalues of the Hessian matrix. Minimization of the P-quartz structure does not result in a real energy minimum as one negative eigenvalue is obtained for the Hessian matrix, indicating the geometry being a saddle point on the potential-energy surface. The fact that P-quartz does not change into cr-quartz during the energy minimization is inher- ent in the Newton-Raphson method, thereby preserving the symmetry of the system. Thus in our force field /?-quartz is not a stable structure at 0 K. However, this does not imply that /?-quartz is not an energy minimum at higher tem- peratures and/or larger unit-cell volumes (Fig. 1).

By applying a constraint on the unit-cell volume of the quartz structure the heat of formation, the vibrational fre- quencies and the structural parameters have been calculated as a function of the constrained volume. Each step in volume change is followed by energy minimization. After energy minimization all interatomic forces are in equilibrium. With increasing unit-cell volume the structural parameters of the a-quartz structure show a rapid change towards the P-quartz values. The P-quartz symmetry (u = 0.5) is achieved at a unit- cell volume & = 0.1208 nm3, the transition volume. Fig. 2 shows the positional parameter u of the silicon atoms as a function of the volume. At the transition volume this param- eter reaches the P-quartz value u = 0.5.

Energy minimization was no longer possible for unit-cell volumes larger than 0.126 nm3, indicating that the quartz structure is no longer stable. Energy minimization was also not possible for unit-cell volumes between 0.1202 and 0.1204 nm3 with P-quartz symmetry, just below the transition volume. The latter could well be caused by the potential surface being not well defined in that region. Unit-cell volumes less then 0.1 172 nm3 result for the P-quartz structure in three negative eigenvalues of the Hessian matrix.

Fig. 3 shows the calculated heat of formation’2 as a func- tion of the unit-cell volume. The figure shows two parabola: the lower one represents structures with a-quartz symmetry, the upper one structures with P-quartz symmetry. The absol- ute energy minimum is the a-quartz structure at a volume V, = 0.1134 nm3. The minimum of the fl-quartz curve is on a higher energy level. At a volume vr = 0.1208 nm3 the two curves intersect and P-quartz is achieved : the transition volume. Each point along the a curve and each point along the P curve right from the transition volume is characterized by a zero gradient and by a Hessian matrix with three zero

P

-7-

Fig. 1 Schematic view of a possible potential surface of the quartz structure and its sensitivity to the unit-cell volume. The two minima of the a-quartz structures come together in one minimum represent- ing the fi-quartz structure at larger unit-cell volumes. The horizontal reaction coordinate is for example the tilt angle’

100 110 120 unit-cell volume/l O3 nm’

Fig. 2 function of the constrained unit-cell volume

Variation of the silicon position in the quartz structure as

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J. CHEM. SOC. FARADAY TRANS., 1992, VOL. 88

-905 r

I - E

z

7 * 0 .- 4-

ti -910-

0

0, z t,

1163

-

120 -9131 ' I ' I ' ' '

100 110 unit-cell volume/l O3 nm3

Fig. 3 Calculated heat of formation of the quartz structure as a function of the constrained unit-cell volume

eigenvalues and no negative eigenvalues. All points shown on the f i curve on the left of the transition volume are character- ized by one negative eigenvalue and hence these are no energy minima. Calculated structural parameters for both quartz structures, @-quartz at the transition volume, are com- pared with experimental data in Table 1. The Si-0-Si angles, both calculated and experimental, are larger in the @-quartz structure than in the structure of a-quartz.

The effect of temperature on the vibrational frequencies of quartz was studied by Nedungadi." The most remarkable feature is shown by the 207 cm-' band which broadens rapidly with increasing temperature with the mean position shifting to lower wavenumbers. This vibration is generally ascribed to the transition of a-quartz to p-quartz. In agree- ment with o thers '~~ we have found that the eigenvector of the Hessian matrix belonging to this mode has almost the same direction as the transition vector between the Q and P-quartz structures. Translation of the atoms of the P-quartz structure along this eigenvector inevitably results in a-quartz upon energy minimization. During our calculations with con- strained volume the frequency of this vibration vanished at the transition volume. For fi-quartz unit cells with a volume less than the transition volume this frequency is imaginary. The relation between the unit-cell volume and this mode and the other a-quartz A, modes, of which three modes become B, modes in the P-quartz region, is shown in Fig. 4. Another

Table 1 Structural parameters of the quartz structures

a-quartz @-quartz

exp. calc. exp. calc.

silicon position

oxygen position U

X

Y

lattice parameters a/nm c/nm

V/nm3 unit-cell volume

density '

Si-0-Si angle angle/degrees

heat of formationZo Af H/kJ mol- ' SiO,

Plg cm-3

0.465

0.415 0.272 0.120

0.49 1 0.540

0.1 130

2.65

144

910.9

0.473

0.418 0.261 0.126

0.494 0.537

0.1134

2.64

147

-912.1

0.5

0.394 0.197 0.167

0.50 1 0.547

0.1189

2.53

155

0.5

0.423 0.21 1 0.167

0.503 0.552

0.1208

2.48

156

-909.6

1000

c I

5 -2 a E 2 500

3

0,

m

0

(Y -

I I I I I I I I I I I I I I I

110 115 120 125

unit-cell volume/l O3 nm3

Fig. 4 Calculated vibrational frequencies of the quartz structure (a- quartz A, symmetry, /?-quartz A, and B, symmetry) as a function of the constrained unit-cell volume

vibration which shows a temperature effect is the a-quartz E mode at 128 cm-'. This Raman (and IR) active mode shifts towards the exciting line as the temperature is raised." At 530°C the wavenumber reaches a value of 98 cm-'. During our calculations the wavenumber in the unstrained a-quartz structure is calculated at 140 cm- ' and shifts downwards to reach the value of 99 cm-' at the transition volume. Fig. 5 shows this mode together with the other E modes of the a- quartz structure with El symmetry for the P structure. At unit-cell volumes below 0.117 nm3 the lowest El mode also vanishes in the region with P-quartz symmetry. Table 2 lists the calculated and experimental vibrational modes of the a and P-quartz structures.

The volume constraint applied in our calculations can be translated into a pressure via the residual force on the lattice parameters. A volume increase is the result of the application of a negative pressure.

Fig. 6 shows the pressure during our calculations as a func- tion of the constrained volume. In Table 3 experimental values for (V, - V)/V, are given as a function of pressure together with our data. Also given are values for the density p of a-quartz at high pressures.

The results of increasing the pressure on the a-quartz unit cell obtained with our molecular mechanics calculations are in agreement with experimental compression results. However, no comparison with experiment is possible for the

L 0

a -2-

5 c 500 > m

O ' i i o ' I 115 I I I 120 I l r l I 125 unit-cell volume/l O3 nm3

Fig. 5 Calculated vibrational frequencies of the quartz structure (a- quartz E symmetry, B-quartz El symmetry) as a function of the con- strained unit-cell volume

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1164 J. CHEM. SOC. FARADAY TRANS., 1992, VOL. 88

Table 2 Experimental and calculated (wavevector k = 0) vibra- tional modes (cm-') of the quartz structures

wavenumber/cm - wavenumber/cm -

symmetry calc. exp. symmetry calc. exp.

A1

A2

E

a-quartz 54 1 464 200 207 329 356 123 1082 495 49 5 105 1080 371 364 798 778 140 128 423 394 708 795

1097 1072 29 5 265 465 450 672 697

1196 1162

P-quartz A, 533 464 Bl 0 inactive

317 inactive 1143 inactive 427

1117 B2 422 inactive

800 inactive

422 409 687 788

1112 1067 E2 29 3 245

455 428 663 688

1219 1173

. . .

. . . A2

El 99 99

101 \

unit-cell volurne/l O3 nm3

Fig. 6 strained unit-cell volume

Pressure on the quartz structure as a function of the con-

Table 3 for a-quartz compared with calculated data

Experimental data on the pressure-volume/density relation

K O - WVO P/g cm -

p/GPa exp.I9 calc. exp.'' calc.

0.5 1 .o 1.5 2.0 2.5 3.0 4.0 0.0 5.0

10.0

0.013 0.009 0.024 0.019 0.035 0.028 0.044 0.037 0.054 0.046 0.063 0.055 0.076 0.07 1

2.65 2.64 2.92 2.89 3.08 3.1 1

negative pressures, the range where a-quartz converts into p- quartz.

Experimental data on the relation between pressure and transition temperature give d?;,/dp = 260 K GPa-'. '7,'8 An increase of 1 GPa in pressure results in the increase in the transition temperature of 260 K. The pressure applied in our calculations at the transition volume is -3 .35 GPa (0 K).

Extrapolation of the experimental transition temperature- pressure relation to 0 K yields a pressure of -3.25 GPa, which is very close to our calculated pressure. The negative pressure of - 1.5 GPa which Silvi et a1.' applied during their calculations is probably insufficient.

Conclusion From a molecular mechanics point of view a-quartz is the stable structure at no-constraint conditions (0 K, 0 Pa, V, = 0.1134 nm3). Applying a negative pressure (by means of a volume constraint) on the structure of a-quartz this structure is converted via a displacive transition into /?-quartz (0 K, -3.35 GPa, V;, = 0.1208 nm3). As our calculations are effec- tively performed at 0 K no distinction can be made regarding the classification of the transition as first or second order.

The volume-pressure curve is in agreement with experi- mental high-pressure data. The negative pressure applied at the transition volume is in agreement with the relation between pressure and transition temperature extrapolated to 0 K.

E.V.B. thanks Akzo Chemicals Amsterdam for financial support.

References 1

2

3 4

5 6

7

8 9

10 11 12

13

14

15

16

17

18

19

20

N. N. Greenwood and A. Earnshaw, Chemistry of the Elements, Pergamon, Oxford, 1984, p. 394. R. W. G. Wyckoff, Crystal Structures, John Wiley, New York, 2nd edn., 1963, vol. 1, p. 312. G. Dolino, Advances in Physical Geochemistry, 1988,7, 17. S . Tsuneyuki, H. Aoki, M. Tsukada and Y. Matsui, Phys. Rev. Lett., 1990, 64, 776. B. Silvi, P. DArco and M. Causa, J. Chem. Phys., 1990,93, 7225. B. D. Saksena and H. Narain, Proc. Ind. Acad. Sci., 1949, A30, 128. J. Etchepare, M. Merian and L. Smetankine, J. Chem. Phys., 1974,60,1873. D. A. Kleinman and W. G. Spitzer, Phys. Rev., 1962, 125, 16. M. M. Elcombe, Proc. Phys. SOC. London, 1967,91,947. T. M. Nedungadi, Proc. Indian Acad. Sci., 1940, Al l , 86. J. F. Scott, Phys. Rev. Lett., 1968, 13, 907. E. de Vos Burchart, V. A. Verheij, H. van Bekkum and B. van de Graaf, Zeolites, in the press. N. L. Allinger, Y. H. Yuh and J-H. Lii, J . Am. Chem. SOC., 1989, 111,8551. E. de Vos Burchart, B. van der Linden, H. van Bekkum and B. van de Graaf, Collect. Czech. Chem. Commun., in the press. B. van de Graaf, J. M. A. Baas and A. van Veen, Recl. Trav. Chim. Pays-Bas, 1980,99,175. B. van de Graaf and J. M. A. Baas, J. Comput. Chern., 1984, 5, 314. L. H. Cohen, W. Klement Jr. and H. G. Adams, Am. Mineral., 1974,59, 1099. M. S. Ghiorso, I. S. E. Carmichael and L. K. Moret, Contrib. Mineral. Petrol., 1979,68, 307. S . P. Clark, Handbook of Physical Constants, The Geological Society of America, Rev. edn., 1966, p. 138, 158. R. C. Weast, CRC Handbook of Chemistry and Physics, CRC Press Inc., Boca Raton, 63rd edn., 1982, D-85.

Paper 1/05868A; Received 19th November, 1991

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