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  1. 1. A molecular dynamics study of ambient and high pressure phases of silica: Structure and enthalpy variation with molar volume Chitra Rajappa, S. Bhuvaneshwari Sringeri, Yashonath Subramanian, and J. Gopalakrishnan Citation: The Journal of Chemical Physics 140, 244512 (2014); doi: 10.1063/1.4885141 View online: View Table of Contents: Published by the AIP Publishing Articles you may be interested in Reactive molecular dynamics study of Mo-based alloys under high-pressure, high-temperature conditions J. Appl. Phys. 112, 013511 (2012); 10.1063/1.4731793 Observation of high pressure o-GeTe phase at ambient pressure in Si-Te-Ge glasses AIP Advances 2, 012172 (2012); 10.1063/1.3696862 Dissociative melting of ice VII at high pressure J. Chem. Phys. 130, 124514 (2009); 10.1063/1.3100771 Hydrostatic compression and high-pressure elastic constants of coesite silica J. Appl. Phys. 103, 053506 (2008); 10.1063/1.2888558 Theoretical study of the molecular motion of liquid water under high pressure J. Chem. Phys. 119, 1021 (2003); 10.1063/1.1578624 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Fri, 04 Sep 2015 07:33:02
  2. 2. THE JOURNAL OF CHEMICAL PHYSICS 140, 244512 (2014) A molecular dynamics study of ambient and high pressure phases of silica: Structure and enthalpy variation with molar volume Chitra Rajappa, S. Bhuvaneshwari Sringeri, Yashonath Subramanian, and J. Gopalakrishnan Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012, India (Received 28 January 2014; accepted 13 June 2014; published online 30 June 2014) Extensive molecular dynamics studies of 13 different silica polymorphs are reported in the isothermal-isobaric ensemble with the Parrinello-Rahman variable shape simulation cell. The van Beest-Kramer-van Santen (BKS) potential is shown to predict lattice parameters for most phases within 2%–3% accuracy, as well as the relative stabilities of different polymorphs in agreement with experiment. Enthalpies of high-density polymorphs – CaCl2-type, α-PbO2-type, and pyrite-type – for which no experimental data are available as yet, are predicted here. Further, the calculated en- thalpies exhibit two distinct regimes as a function of molar volume—for low and medium-density polymorphs, it is almost independent of volume, while for high-pressure phases a steep dependence is seen. A detailed analysis indicates that the increased short-range contributions to enthalpy in the high-density phases arise not only from an increased coordination number of silicon but also shorter Si–O bond lengths. Our results indicate that amorphous phases of silica exhibit better op- timization of short-range interactions than crystalline phases at the same density while the magni- tude of Coulombic contributions is lower in the amorphous phase. © 2014 AIP Publishing LLC. [] I. INTRODUCTION Silica (SiO2) is the most abundant oxide on the earth’s crust. It exists in many different crystalline polymorphs with SiO4 tetrahedra as the principal structural unit, except for high-pressure forms in which SiO6 octahedra are observed. The polymorphs differ only in the connectivity of the basic structural units and hence are quite close in energy. The struc- tural complexity and subtle energy differences between dif- ferent silica polymorphs make the modelling of these systems non-trivial. Many accurate forcefields have been developed to model silica polymorphs.1–15 Several workers have investigated different silica phases using a variety of computational techniques. Tsuneyuki et al.16 carried out molecular dynamics (MD) simulations to show that α-cristobalite transforms to an orthorhombic phase at around 16.5 GPa, which then transforms to stishovite at 23 GPa. Tse and Klug17 performed molecular dynamics sim- ulations using the van Beest-Kramer-van Santen (BKS) po- tential to calculate static and dynamical properties of several silica polymorphs. They find that the equilibrium structures, elastic constants as well as the calculated vibrational spectra agree well with experimental values. Keskar and Chelikowsky18 calculated the thermody- namic properties of quartz, coesite, and stishovite from the phonon density of states using the interatomic potentials de- rived by Tsuneyuki et al.1 Bourova and coworkers19 have investigated the high-temperature structure and dynamics of coesite using lattice dynamics and molecular dynamics sim- ulations. The high-pressure behaviour of α-cristobalite has been investigated by Garg and Sharma20 by molecular dy- namics simulations using the BKS and Tsuneyuki-Tsukada- Aoki-Matsui (TTAM) potentials. Poole and coworkers21 and Herzbach et al.22 have car- ried out simulation studies to understand the phase diagram of silica polymorphs, structural phase transformations, and ther- mochemical properties of silica using three different poten- tials. Huang and coworkers23,24 as well as Liang et al.25 have both investigated the mechanism of phase transformations be- tween different phases of silica. In recent times there is increased interest in ab initio studies of silica.26–29 Durandurdu has investigated the effect of anisotropic stress on α-cristobalite by means of constant pressure ab initio simulations. He observed the formation of several structures: anatase-like, stishovite, and CaCl2-type structures.26 In order to understand the energy dependence on TOT angle ab initio simulations were carried out on silica and germania crystalline systems. They found that SiOSi angle is nearly flat over a wide range of angles whereas this was not the case with GeOGe angle.27 Shi et al.28 explored the effect of temperature on ring diameters in zeolites sodalite, ferrierite and silicalite to understand negative thermal expansion coef- ficients as well as variation in ring diameter with temperature using both ab initio and classical simulations. Continuous ran- dom network model of an amorphous glass has been studied using ab initio approaches and the short-range order has been compared with that in crystalline α-quartz.29 There are several ab initio studies investigating band structure, and the phase boundary of α-quartz and other high pressure phases up to stishovite with the help of Hartree- Fock and density functional theory (DFT) calculations. The effect of Hamiltonian and basis set on the vibrational fre- quency of α-quartz has also been studied. Ab initio studies have also been used to calculate elastic properties as well as investigate the surface structure of α-quartz and high pressure silica.12–15,31 0021-9606/2014/140(24)/244512/11/$30.00 © 2014 AIP Publishing LLC140, 244512-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Fri, 04 Sep 2015 07:33:02
  3. 3. 244512-2 Rajappa et al. J. Chem. Phys. 140, 244512 (2014) coesite CaCl2−typestishovit e alpha−PbO2−type pyrite−type Cristobalite Tridymitealpha−quartz ZSM−5 ZSM−11 Zeolite Beta Faujasite EMT (a) (b) (c) FIG. 1. Various polymorphs of silica investigated in this study are shown: (a) high pressure phases; (b) medium density phases at ambient temperature and pressure; and (c) porous phases of silica. Zwijnenburg et al.32 have compared experimental ther- mochemical data for silica polymorphs with simulation studies using different force fields and with DFT calculations. More recently, Liang and coworkers33 have reported molec- ular dynamics studies of the structural transitions and phase transformations in melanophlogite, a low-pressure silica polymorph. Experimental investigations of the thermodynamic prop- erties of silica polymorphs are well-known. Johnson and coworkers have studied the thermodynamic properties of silicalite.34 Petrovic et al. have measured the enthalpy of for- mation of a number of high-silica zeolites using high temper- ature solution calorimetry and drop calorimetry.35 They find that the enthalpy of the high-silica zeolitic forms are about 7–14 kJ/mol higher than the enthalpy of quartz.35,36 Spec- troscopic (IR, Raman) studies of silica polymorphs are also available.37,38 Navrotsky et al. have measured the heat of for- mation of various phases of high-silica zeolites as well as some dense forms of silica as a function of molar volume.39 They have also reported the experimentally measured entropy of formation for some silica polymorphs.39 Figure 1 shows the different silica polymorphs inves- tigated here. Coesite, stishovite, CaCl2-type, α-PbO2-type, and pyrite-type silica are high-pressure polymorphs. α-quartz, cristobalite, and tridymite are medium-density polymorphs, while ZSM-5, ZSM-11, zeolite beta, faujasite, and EMT (hexagonal faujasite) are low-density or porous silica poly- morphs. In spite of innumerable studies, there is lack of a com- prehensive and comparative study of the low, intermediate and high density polymorphs of silica. Is there a single in- termolecular potential that can predict the complexity of This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Fri, 04 Sep 2015 07:33:02
  4. 4. 244512-3 Rajappa et al. J. Chem. Phys. 140, 244512 (2014) the phase diagram and the innumerable structures of silica correctly? How well is the thermodynamics of silica phases at various densities reproduced? In the present study we have chosen to use the BKS po- tential to model 13 different phases of silica polymorphs over a range of densities since (i) it has a simple form, (ii) we focus on structural properties in this study, and the BKS potential is known to model structural and thermodynamic properties of even high-pressure phases quite well.17,22 BKS potential is known to have certain limitations— many-body terms are not accounted for, polarization effects are not factored in, and there is an implicit assumption that there are no significant changes in electronic structure un- der different temperatures and pressures.7 Also, vibrational properties estimated using BKS are known to be only qual- itatively in agreement with experiment.17 A major drawback of the BKS model is that the dispersive term coefficients are unphysically large and the partial charges used are too low.40 By far the most serious shortcoming of BKS is that it leads to both coesite and stishovite being more stable than α-quartz— this can probably be attributed to the large dispersive term in the potential.22,40 This is contrary to experiment as well as DFT studies (in the generalized gradient approximation) which predict α-quartz to be the most stable polymorph.12 BKS is also known to overestimate the transition pressure for the stishovite I → II transition.22 We focus on the structure and thermodynamic properties of SiO2 polymorphs and the results are compared with the experimental data. Simulation of silica glass is also reported and provides interesting insights into its metastability and its origin. Finally, metastability of silica crystalline polymorphs is briefly discussed. II. METHODS A. Molecular dynamics Molecular dynamics simulations were performed in the NPT ensemble using a variable shape simulation cell. The simulation cell is represented by three cell vectors, a, b, and c, following Parrinello-Rahman.41 Unlike in the original formu- lation, we have restricted the degrees of freedom associated with the cell matrix, h = (a,b,c) to six by choosing a to be along the x-axis, b to be in the xy-plane, and c in any direc- tion. This prevents the rotation of the simulation cell observed when the cell matrix has nine degrees of freedom.42 B. Intermolecular potential The BKS potential was used to model all the silica polymorphs:3 φαβ(r) = Aαβexp[−Bαβr] − Cαβ/r6 + qαqβe2 /r, (1) with α, β = Si, O. r is the distance between the two ions of types α and β. Charges of +2.4 and −1.2 are assigned to the Si and O atoms, respectively. The coefficients, Aαβ, Bαβ, and Cαβ of the Buckingham potential are given in Table I. The BKS potential is known to exhibit an unphysical di- vergence to −∞ at small distances. This could be a problem TABLE I. Intermolecular interaction potential for molecular dynamics sim- ulation of silica polymorphs. Interaction Aαβ B−1 αβ Cαβ type (kJ/mol) Å (kJ/mol)Å6 O–O 1.339962d+05 0.362319 1.688493d+04 Si–O 1.737098d+06 0.205205 1.288446d+04 especially while simulating high pressure phases (pyrite-type and α-PbO2-type silica) or while simulating an amorphous phase at high temperature before quenching it. To circumvent this, MD simulations were carried out on such phases using the modified BKS potential (i.e., the original BKS potential with an additional 30-6 Lennard-Jones-type term):43 φαβ(r) = φBKS αβ (r) + 4 αβ σαβ rαβ 30 − σαβ rαβ 6 , (2) where φBKS αβ (r) is the original BKS potential and rαβ is the separation between atoms α and β, which may be either sili- con or oxygen atoms. Si–Si and σSi–Si are both set to zero. Si–O and O–O are taken to be 0.2988995 and 0.10140858 kJ/mol, respectively. σSi–O and σ−O–O are given the values 1.3136 and 1.7792 Å, respectively.43 The additional term does not alter the form of the original BKS potential at large separations, but effectively prevents the negative divergence at short distances. This form of the mod- ified BKS potential has been used by several groups, specifi- cally while simulating silica at high temperatures and/or high pressures.44–46 Several other modifications have also been proposed to correct for the unphysical divergence at short distances.47–52 In the classical molecular dynamics simulations whose results we report here, the total interaction energy has contri- butions from atom-atom (Si–O, O–O, Si–Si) repulsive terms which are always positive. The contribution for this comes from the first term on the right hand side of Eq. (1). The sec- ond contribution is the attractive dispersion interaction term which is always negative. The sum of these two terms is what is termed by us here as the van der Waals interaction energy, Evdw. This is also referred to as short-range interaction en- ergy. The electrostatic interaction due to charges on the atoms gives rise to ECoul. This is also termed as long-range interac- tion energy. The total interaction energy is Epot = Evdw + ECoul, (3) where Evdw = Erepul + Edisp = α β (Aαβexp(−Bαβr) − Cαβ/r6 ). (4) There is an additional contribution arising from the PV term: H = Epot + PV. (5) At high pressures, the PV term is significant. Ab initio calculations report energies which are typically of the order12 of around −23 eV. Ab initio energies include electronic contributions, zero-point energy of optical modes, thermal This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Fri, 04 Sep 2015 07:33:02
  5. 5. 244512-4 Rajappa et al. J. Chem. Phys. 140, 244512 (2014) corrections, etc., while empirical potentials include disper- sion (fluctuating dipole-fluctuating dipole moment, and other higher order terms), repulsion term (due to Pauli exclusion principle arising from electron cloud overlap) and long-range electrostatic contributions. It is therefore difficult to com- pare the ab initio energies with the energies from empiri- cal calculations. However, the differences in energies, for ex- ample, between silica polymorphs should be comparable in magnitude. III. COMPUTATIONAL DETAILS The DLPOLY set of programs53 was used to carry out molecular dynamics simulations at room temperature in the NPT ensemble using a variable shape simulation cell for each of the silica polymorphs. The simulation cell size was cho- sen sufficiently large so that a cut-off radius of 12 Å could be used in the calculation of short-range interactions. In order to get good statistics, larger system sizes were used for the high- pressure phases. For these, a larger cut-off of 14 Å has been employed. The simulation cell consisted of 192 SiO2 units (for low density faujasite) to 2916 SiO2 units (for high den- sity pyrite-type silica). Table II lists the system size used for each of the polymorphs studied here. Note that simulations have been carried out on large systems and for reasonably long durations. For high pressure phases simulations have been carried out on even larger systems. All simulations were performed so that the conservation of energy is better than 1 in 104 . Two different glassy phases of silica were also simulated, each one starting from the crystalline structure of α-quartz and tridymite. We label these Glass-A and Glass-T, respec- tively. The modified BKS potential was used in these simu- lations. The simulation cell consisted of 882 SiO2 units and 2016 SiO2 units for simulating Glass-A and Glass-T, respec- tively (see the supplementary material54 for details of prepa- ration of the amorphous silica phases). Periodic boundary conditions were employed. Electro- static interactions were evaluated using the Ewald summation technique. A time step of 1 fs was used for simulating most of the crystalline polymorphs of silica. For the high-pressure phases – stishovite, CaCl2-type, α-PbO2-type, and pyrite-type – the timestep had to be reduced to 0.5 fs in order to obtain good energy conservation. The durations of the equilibration and production runs are also listed in Table II. IV. RESULTS AND DISCUSSION A. Structure Although calculations on silica polymorphs have been re- ported previously, they have been restricted to low density polymorphs.9 This is the first study which reports results of both low, intermediate, and high density polymorphs using a single potential. In general many potentials include terms such as bonded term, three-body angle dependent, and finally four-body dihedral angle terms. However, it is well-known that the existence of many different phases of silica arises from the possibility to explore a large range of angles in the three-body term. Hence, it would be difficult for a potential with three-body terms to account for such a large number of phases. BKS potential, on the other hand, prescribes a simple atom-atom “non-bonded” interaction which, as we shall see, can account for the extraordinarily wide range of structures that silica can exist in. Table III shows the average values of the lattice param- eters obtained from our variable shape NPT ensemble simu- lations for each silica polymorph. The corresponding exper- imental values are also indicated (see caption to Table III). Clearly, the BKS forcefield (which we have used in this study) is able to correctly model lattice parameters across the whole spectrum of densities and molar volumes. It is encouraging to see that in most cases, the deviation from the experimental value is less than 5%. Although the BKS potential was primar- ily parametrized for tetra-coordinated silica,3 we find that this TABLE II. The simulation cell consists of na, nb, and nc unit cells along each of a, b, and c directions. In addition, the number of SiO2 units in the simulation cell is tabulated for each polymorph. Silica No. of Structure Run length polymorph na nb nc SiO2 units reference Equilibration Production Pyrite-type 9 9 9 2916 55 1 ns 750 ps α-PbO2-type 9 7 8 2016 56 1 ns 1 ns CaCl2-type 8 8 15 1920 56 1 ns 1 ns Stishovite 8 8 15 1920 57 1 ns 1 ns Coesite 5 3 9 2160 58 1 ns 1 ns Glass-A 7 7 6 882 250 ps 250 ps α-quartz 7 7 6 882 59 1 ns 1 ns Cristobalite 6 6 4 576 60 250 ps 250 ps Glass-T 2 7 3 2016 250 ps 250 ps Tridymite 2 7 3 2016 61 250 ps 250 ps ZSM-5 2 2 2 768 62 1 ns 1 ns ZSM-11 2 2 2 768 63 250 ps 250 ps Zeolite beta 2 2 1 256 64 250 ps 250 ps Faujasite 1 1 1 192 65 250 ps 250 ps EMT 2 2 2 768 66 250 ps 250 ps This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Fri, 04 Sep 2015 07:33:02
  6. 6. 244512-5 Rajappa et al. J. Chem. Phys. 140, 244512 (2014) TABLE III. The average values of simulation cell volume, VMD , as well as the average unit cell parameters— a , b , c , α, β, and γ as obtained from the variable shape MD simulations are tabulated for each silica polymorph. For the unit cell edge lengths and molar volume, the percentage deviation with respect to experimental values for the crystalline form is indicated in parentheses. For the angles between unit cell axes, the experimental value for the crystalline form is indicated in parentheses, when it is different from the experimental value. Silica VMD a b c α β γ polymorph (Å3) (Å) (Å) (Å) (deg.) (deg.) (deg.) Pyrite-type 44 758.0 (1.2) 4.048 (3.0) 3.881 (−1.3) 3.909 (−0.5) 90.0 90.0 90.0 α-PbO2-type 37 835.4 (4.6) 3.845 (3.6) 4.644 (−0.2) 4.204 (1.1) 90.0 90.0 90.0 CaCl2-type 36 350.1 (4.7) 3.861 (1.4) 3.861 (1.2) 2.540 (2.0) 90.0 90.0 90.0 Stishovite 43 026.3 (−3.6) 4.112 (−1.6) 4.112 (−1.6) 2.651 (−0.5) 90.0 90.0 90.0 Coesite 73 790.9 (−1.7) 7.001 (−1.9) 12.391 (0.2) 7.229 (0.8) 90.0 121.0 (120.0) 90.0 α-quartz 33 224.2 (1.7) 4.938 (0.5) 4.938 (0.5) 5.443 (0.7) 90.0 90.0 120.0 Cristobalite 24 639.0 (−7.6) 4.911 (−1.2) 4.911 (−1.2) 6.557 (−5.3) 90.0 90.0 90.0 Tridymite 89 170.6 (−10.5) 18.087 (−2.4) 4.688(−6.3) 23.268(−2.3) 90.0 105.6 (105.82) 90.0 ZSM-5 42 621.1 (5.0) 20.263 (1.1) 20.256 (1.9) 13.629 (1.9) 90.0 90.0 90.0 ZSM-11 43 203.2 (4.6) 20.332 (1.3) 20.333 (1.3) 13.660 (1.9) 90.0 90.0 90.0 Zeolite beta 16 931.5 (1.7) 12.679 (0.2) 12.679 (0.2) 26.769 (1.4) 90.0 90.0 90.0 Faujasite 14 273.9 (6.1) 24.739 (2.0) 24.739 (2.0) 24.739 (2.0) 90.0 90.0 90.0 EMT 57 658.4 (5.1) 17.496 (1.6) 17.496 (1.6) 28.568 (1.7) 90.0 90.0 120.0 forcefield yields reasonable results for six-coordinated high- pressure phases as well. This is in agreement with the work of Tse and others.17,67 We believe the atom-atom potential form is responsible for the wonderful ability of BKS potential to account for polymorphism in silica. B. Thermodynamics 1. Metastability of silica polymorphs Table IV lists the average temperature, pressure, and mo- lar volumes obtained from the variable shape NPT ensem- ble simulations for all the silica polymorphs. Also indicated are the values of the enthalpies and the total energies of in- teraction, along with the short-range and long-range energy components. We note that the Evdw is the van der Waals con- tribution which is the sum of the Edisp and Erepul. The former (dispersion term) is always negative and favorable while the latter (repulsive) is always positive and destabilizes. As we can see, the sum of these two terms is negative and favor- able up to stishovite but for still higher pressure polymorphs, the energy is unfavorable. The error bars are also reported in Table IV. The errors are small suggesting that the obtained values are reliable. Experimentally, α-quartz is known to be the most sta- ble silica polymorph at ambient conditions. We find that at TABLE IV. The total energy of interaction, as well as the short-range and long-range contributions are tabulated for all the silica polymorphs simulated in this study (variable shape MD simulation), as a function of molar volume. The average volume, temperature, and pressure obtained from the simulation are indicated along with the set values in parentheses. Values reported are averaged over 250 ps, except for those polymorphs marked with an asterisk for which the averages are over a period of 1 ns. Silica V T P Evdw ECoul Epot H polymorph (cm3/mol) (K) (katm) (kJ/mol) (kJ/mol) (kJ/mol) (kJ/mol) Pyrite-type∗ 9.24(9.14) 299.7(300) 2674.8(2674.8) +827.30(1.73) − 5941.30(0.82) − 5114.00(1.09) − 2597.64(4.96) α-PbO2-type∗ 11.30(10.81) 299.0(300) 1188.1(1188.1) +258.06(1.54) − 5761.90(0.93) − 5503.97(0.71) − 4132.19(4.74) CaCl2-type∗ 11.40(10.89) 298.3(300) 1188.2(1188.1) +256.22(11.7) − 5781.77(6.85) − 5525.52(4.90) − 4142.03(36.2) Stishovite∗ 13.49(14.02) 299.2(300) 99.1(99.0) − 167.54(5.5) − 5475.52(5.1) − 5643.23(0.48) − 5496.35(19.9) Coesite∗ 20.57(20.58) 297.5(300) 44.4(44.4) − 79.60(1.93) − 5555.09(1.78) − 5634.72(0.25) − 5532.87(6.44) Glass-A 22.68(22.68) 299.7(300) 0.0(0.0) − 93.45(1.54) − 5481.07(1.63) − 5574.49(6.07) − 5563.38(4.99) α-quartz∗ 22.68(22.71) 298.1(300) 0.0(0.0) − 90.25(1.38) − 5533.79(1.38) − 5624.04(0.17) − 5612.93(3.75) Cristobalite 25.76(25.49) 288.6(298) 0.1(0.0) − 87.08(2.26) − 5520.49(2.32) − 5607.64(0.93) − 5596.70(12.2) Glass-T 26.64(23.87) 297.6(300) 0.0(0.0) − 86.50(1.45) − 5478.67(1.98) − 5564.98(1.63) − 5554.07(6.46) Tridymite 26.64(27.52) 290.5(298) 0.2(0.0) − 85.56(1.82) − 5524.80(2.05) − 5610.12(1.31) − 5599.21(16.3) Glass (Expt.) 27.27 ZSM-5∗ 33.42(33.65) 298.1(298) 0.0(0.0) − 55.46(4.32) − 5529.30(1.51) − 5584.77(0.27) − 5573.57(4.32) ZSM-11 33.88(34.02) 373.2(373) 0.0(0.0) − 55.27(1.78) − 5525.13(1.79) − 5580.34(0.22) − 5566.41(5.15) Zeolite beta 39.83(38.56) 298.3(298) 0.0(0.0) − 53.99(2.53) − 5518.36(2.54) − 5572.66(0.23) − 5561.33(7.19) Faujasite 44.77(44.29) 298.8(298) 0.0(0.0) − 51.37(4.11) − 5504.69(4.12) − 5556.25(0.32) − 5544.79(12.0) EMT 45.21(46.51) 293.0(293) 0.0(0.0) − 51.32(2.72) − 5504.82(2.72) − 5556.25(0.19) − 5545.18(8.02) Stishovite∗ 13.81(14.02) 299.7(300) 0.0(0.0) − 207.27(3.6) − 5436.98(3.6) − 5644.27(0.09) − 5632.81(12.5) This article is copyrighted as indicated in the article. 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  7. 7. 244512-6 Rajappa et al. J. Chem. Phys. 140, 244512 (2014) ambient conditions (300 K and 1 atm pressure), the BKS po- tential indicates that stishovite is the most stable phase and is marginally lower in energy (by about 20 kJ/mol) than α- quartz. Poole and coworkers also found that stishovite is more stable than α-quartz at 300 K and 1 atm in their simulation study of the phase diagram using BKS potential.21 Thus, it is clear that BKS does not predict the order of stability correctly at room temperature and pressure. Our NVE simulations of stishovite and α-quartz (data not shown here) also indicate that the BKS potential favours the stishovite phase at ambient conditions. However, under the experimental conditions at which each of these phases is known to exist, the order of sta- bility could be different. Hence, we carried out simula- tions at the corresponding pressure and temperature to see what order of stability is indicated. The results from the variable shape Parrinello-Rahman isothermal-isobaric sim- ulations performed at different pressures and temperatures showed significant changes in the order of stability. Under these conditions, we find that α-quartz is the most stable phase. The enthalpies of various phases at their respective pressures and temperatures are listed in Table IV. Note that α-quartz is the most stable phase by about 117 kJ/mol as com- pared to stishovite. We also attempted to study a few phases which share a co-existence line in the phase diagram as a function of pres- sure. We have shown in Figure 2(a) the variation in enthalpy as a function of pressure for coesite and α-quartz. In the case of coesite, the pressure was decreased (the arrow indicates the direction of pressure change) from 44 katm to 1 atm, while in the case of α-quartz the pressure was increased from 1 atm to 100 katm. Note that the lines do not intersect in the range of pressures investigated. However, this is not surprising since the hysteresis associated with the first-order transitions will ensure that the transitions actually take place at much higher or lower pressures. In the case of stishovite-α-quartz (shown in Figure 2(b)), the two come close to intersection but this appears to be at some negative pressure. More detailed stud- ies are required for understanding phase transitions in silica polymorphs and these are under progress. 0 20 40 60 80 100 -5700 -5650 -5600 -5550 -5500 -5450 -5400 H(kJ/mol) α-quartz coesite 0 20 40 60 80 100 P (katm) -5700 -5650 -5600 -5550 -5500 -5450 -5400 H(kJ/mol) α-quartz stishovite (a) (b) > < > < FIG. 2. Variation of enthalpy with pressure for α-quartz, coesite, and stishovite. 0 10 20 30 40 50 -300 0 300 600 900 Evdw (kJ/mol) 0 10 20 30 40 50 -6000 -5800 -5600 -5400 ECoul (kJ/mol) 0 10 20 30 40 50 V (cm 3 /mol) -5800 -5600 -5400 -5200 -5000 Epot (kJ/mol) 1 2, 3 4 5 6 7, 8 9, 10 11 12, 13 FIG. 3. Plot of total intermolecular interaction energy, Epot, short-range van der Waals interaction energy, Evdw, and Coulombic energy contri- bution, ECoul obtained from molecular dynamics simulations in the vari- able shape NPT ensemble at the temperatures and pressures indicated in Table IV. The following polymorphs are shown in the figure: (1) pyrite-type silica, (2) α-PbO2-type silica, (3) CaCl2-type silica, (4) stishovite, (5) coesite, (6) α-quartz, (7) cristobalite, (8) tridymite, (9) ZSM-5, (10) ZSM-11, (11) ze- olite beta, (12) faujasite, and (13) EMT. The dotted lines are high-order fitted polynomials and serve to guide the eye. Filled symbols represent the corre- sponding values for Glass-A and Glass-T. Error bars are also indicated for all quantities. In all cases, the error bars are smaller than the size of the symbol used. 2. Energy contributions as a function of molar volume Previous calculations suggest that the differences in the enthalpies between large pore zeolites are small.9 They fur- ther suggest that the short-range contributions are also not very different for these zeolites. In order to understand the metastability of the various polymorphs that we have simu- lated in the present study, we have listed the short-range and the long-range contributions in Table IV. Further, Figure 3 shows the variation of the total interaction energy and the as- sociated short-range and long-range components as a function of molar volume for all the polymorphs studied here. The values obtained in the present study and previ- ous calculations show good agreement with each other. For crystalline cristobalite, the Coulombic contribution esti- mated from this MD study is −5520 kJ/mol. This compares well with values of −5525 kJ/mol and −5530 kJ/mol, ob- tained from simulation studies by Yamahara et al.68 and Li et al.,69 respectively, for BKS potential. The values reported by Li and coworkers69 for the Coulombic components for α- quartz, tridymite, ZSM-11, and zeolite beta are −5540.53, −5547.72, −5537.16, and −5526.54 kJ/mol which are com- parable to our values of −5533.79, −5524.80, −5525.13, and −5518.36 kJ/mol, respectively. The maximum difference is for tridymite which is around 0.4%. Before we discuss the metastability of the various crys- talline polymorphs, it is interesting to understand the differ- ence between the crystalline and the amorphous forms of silica. There have been a few ab initio studies which have investigated pressure-induced amorphization of silica.10,30,70 The short-range contributions to the total energy is slightly This article is copyrighted as indicated in the article. 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  8. 8. 244512-7 Rajappa et al. J. Chem. Phys. 140, 244512 (2014) 0 30 60 90 120 150 180 0 0.003 0.006 0.009 0.012 0.015 f(θ) crystal glass 0 30 60 90 120 150 180 0 0.002 0.004 0.006 0.008 0 30 60 90 120 150 180 θ (degrees) 0 0.002 0.004 0.006 0.008 f(θ) 0 30 60 90 120 150 180 θ (degrees) 0 0.001 0.002 0.003 0.004 0.005 α-quartz tridymite O-Si-O Si-O-Si O-Si-O Si-O-Si FIG. 4. The distribution of different angles is shown for the glass as well as well as the crystal of comparable density. larger in magnitude for the glassy phase as compared to the crystalline polymorph with similar molar volume. This means that the amorphous phases have a lower short-range energy as compared to the crystalline form at the same density! In other words the glasses optimize the short-range interac- tions better than their crystalline counterparts. Thus, Glass-A and Glass-T are found to have short-range interaction ener- gies of −93.45 kJ/mol and −86.50 kJ/mol, while the corre- sponding crystalline phases – α-quartz and tridymite – are associated with short-range energies of −90.25 kJ/mol and −85.56 kJ/mol, respectively. The reason for better optimization of the short-range en- ergy in the case of glasses can be understood if we look at the structure of the glass. In the glassy phase, it is seen that although the tetrahedra remain intact, the interconnec- tion between two tetrahedra are altered significantly. As is evident from Figure 4 the distribution of Si–O–Si angles is altered significantly. The distribution changes from bimodal to monomodal and also extends over a much larger range of angles. This is consistent with what is known of the “soft” bending potential at the oxygen atom; i.e., the energetic cost associated with distorting the Si–O–Si angle is very small.71 This suggests that the link between tetrahedral units is much more flexible and provides for much larger choice. This is not the case for O–Si–O angle which is broader for the glass as compared to the crystal. From the figure it is also evident that in tridymite much smaller angles are seen as compared to α-quartz. We see that better optimization and larger flexibil- ity and existence of short-range order leads to lower energies (higher in magnitude) for the glassy phase. In contrast, the Coulombic contribution is higher (lower in magnitude) for the glassy phase. The crystalline polymorph has a significantly lower Coulombic energy as compared to the glassy phase of similar density. This is understandable since the absence of long range order in the glassy phase leads to lower contributions from the Coulombic component of the interaction energy (see Table IV for the actual values). This trend in the relative values of the Coulombic and short-range contributions between the amorphous and the crystalline forms is similar to that observed by Yamahara et al.68 in their MD studies of crystalline and amorphous cristobalite. They find that the Coulombic component of in- teraction energy for the crystal is about 25 kJ/mol more favourable than that for the amorphous cristobalite. In the present study, we find that the Coulombic contributions for crystalline α-quartz and crystalline tridymite are about 52.7 kJ/mol and 46.1 kJ/mol lower (i.e., more favourable) than the respective amorphous phases. Yamahara et al. have used a different silica potential and so only a qualitative comparison is possible. Two distinct regimes can be seen in the variation of the enthalpy and other quantities with molar volume—a sharp de- pendence at high-pressures (for stishovite and beyond) and a very slight variation for lower density polymorphs. Note that stishovite and still higher pressure phases all are six- coordinated. Thus, it appears that the change from weak de- pendence to strong dependence on molar volume is associated with the change from 4-coordinated to 6-coordinated phases. This is an important link between the thermodynamics of sil- ica phases and the structure. The present finding suggests that there is a thermodynamic signature of the change in coordina- tion of silicon. This has not been observed previously, to the best of our knowledge. The underlying reason for this marked change in the dependence of Epot, the total potential energy of interaction with molar volume becomes clear if we look at Table IV and Figure 3. From Figure 3 it is evident that the variations with molar volume of Epot and Evdw are similar. From Table IV, we see that at high pressures, the contribution of the short- range interaction energy, Evdw, to the total energy becomes significant. Thus, the steep variation of Epot with molar vol- ume, V , seen at high pressures, is due to significant increase in the short-range energy with decrease in molar volume. The Coulombic contribution, on the other hand, does not exhibit any such noticeable change. This is to be expected since there is no significant change in long-range order. Thus, at higher densities, the short-range interaction energy has a stronger de- pendence on molar volume than at smaller pressures. To understand the reasons for the sudden change in the short-range interaction energy with molar volume, we have further analysed the nearest neighbour structure in detail. Figure 5 shows the dependence of the BKS potential for the Si–O interaction on the Si–O distance. The sharp increase in the contribution from the short-range interaction can be understood in terms of the nearest neighbour structure. In α-quartz, we see that there are just four oxygens at a dis- tance of 1.61 Å. We see from Figure 5 that energy becomes unfavourable below 1.52 Å. Further, the contribution to the short-range energy rapidly increases with decrease in Si–O distance. In coesite, the Si–O distance and the coordination number of Si are very similar to α-quartz. But as the pressure increases, first there is an increase in the coordination num- ber of silicon. Stishovite exhibits 6 coordination with most of the Si–O bonds longer than in α-quartz (4 at 1.76 Å and 2 at 1.81 Å as obtained from MD simulation). The increased co- ordination leads to larger short-range contribution. With fur- ther increase in pressure, the coordination number remains essentially unchanged but the Si–O bond distance gradually This article is copyrighted as indicated in the article. 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  9. 9. 244512-8 Rajappa et al. J. Chem. Phys. 140, 244512 (2014) 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 r (A) -250 0 250 500 750 1000 1250 U(Si-O)(kJ/mol) BKS potential Modified BKS potential pyrite−type alpha−PbO2−type CaCl2−type alpha−quartz coesite stishovite FIG. 5. The BKS potential for the Si–O interaction is plotted as a function of the Si–O distance. The modified form of the potential is also indicated in the figure. decreases: 4 at 1.64 Å and 2 at 1.62 Å for both CaCl2- and α- PbO2-type. In the highest pressure polymorph – pyrite-type silica – all the six Si–O bonds are of length 1.61 Å. These changes can be more clearly seen in Figure 6 for various high- pressure phases of silica and α-quartz. The changes in coor- dination numbers with distance for various phases are also shown in this figure. Thus, the shorter Si–O bond lengths from stishovite onwards lead to a sharp increase in the contribution from short-range interactions. Thus, the present analysis sug- gests that the increase in the enthalpies of the various high pressure phases arises principally from changes in the first 0 10 20 30 40 50 V (cm 3 /mol) -6000 -5000 -4000 -3000 -2000 H MD (kJ/mol) 0 10 20 30 40 50 -920 -910 -900 -890 -880 -870 -860 -850 ΔH expt (kJ/mol) 1 2,3 4 5 6 7 7 8 9 10 11 12 13 (a) (b) 4 5 6 9 10 11 12 13 FIG. 7. Variation of enthalpy with molar volume for different silica poly- morphs (a) from experimental data, and (b) from variable shape NPT ensem- ble molecular dynamics simulations. The following polymorphs are shown in the figure: (1) pyrite-type silica, (2) α-PbO2-type silica, (3) CaCl2-type silica, (4) stishovite, (5) coesite, (6) α-quartz, (7) cristobalite, (8) tridymite, (9) ZSM-5, (10) ZSM-11, (11) zeolite beta, (12) faujasite, and (13) EMT. The dotted lines are high-order polynomials and are meant to guide the eye. Filled symbols represent the corresponding values for Glass-A and Glass-T. Error bars are indicated for values from MD; in all cases, the error bars are within the size of the symbol used. coordination shell: increase in coordination number as well as shorter Si–O bond lengths. Figure 7 shows (a) the experimental heats of forma- tion from available literature, and (b) the enthalpies obtained from our MD simulations plotted as a function of molar vol- ume for all the polymorphs investigated in this study. There can be no quantitative comparison between the enthalpies of formation obtained from experiment and the enthalpies obtained from simulation. We show the variation of H = H(polymorph) − H(α-quartz) as a function of molar volume from experiment and MD in Figure 8. We can see that while the trends are similar, there are quantitative differences be- tween them. We have not shown the H for polymorphs beyond stishovite as there is no experimental data for high pressure phases beyond stishovite. Experimental data show 1.4 1.5 1.6 1.7 1.8 1.9 2 r (A) 0 5 10 15 20 25 gSi-O (r) α-quartz coesite stishovite CaCl2 -type α-PbO2 -type pyrite-type 1.4 1.5 1.6 1.7 1.8 1.9 2 r (A) 0 1 2 3 4 5 6 7 nSi-O (r) α-quartz coesite stishovite CaCl2 -type α-PbO2 -type pyrite-type FIG. 6. Si–O radial distribution functions and associated coordination numbers are shown for the high-pressure polymorphs studied here. The corresponding quantities for α-quartz have also been plotted for comparison. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Fri, 04 Sep 2015 07:33:02
  10. 10. 244512-9 Rajappa et al. J. Chem. Phys. 140, 244512 (2014) 10 20 30 40 50 V (cm 3 /mol) 0 50 100 150ΔH(kJ/mol) ΔH from exptl data ΔH from MD 4 5 6 7 9 11 8 12,13 10 FIG. 8. Dependence of H (=H(polymorph) − H(α-quartz)) on molar vol- ume from experimental data and from the present variable shape NPT MD study. The following polymorphs are shown in the figure: (4) stishovite, (5) coesite, (6) α-quartz, (7) cristobalite, (8) tridymite, (9) ZSM-5, (10) ZSM-11, (11) zeolite beta, (12) faujasite, and (13) EMT. that the enthalpy of stishovite is considerably higher than that of coesite, suggesting a steeper variation of enthalpy with molar volume at high pressures. Our own MD simu- lations suggest that this is indeed the case. Note, however, that there is a qualitative agreement between the BKS derived curve and the experimental curve suggesting that the order of the enthalpies of the various phases is in agreement with experiment. Further, the additional high pressure phases (CaCl2-type, α-PbO2-type, pyrite-type) studied here show that this steep dependence becomes even steeper with reduction in molar volume. From Table IV one can get an approximate esti- mate of the magnitude of enthalpy difference between these phases and α-quartz. Thermodynamic data from experiments on these high pressure phases are not yet available. The present study thus reports the enthalpies of the high pressure phases of silica for the first time. It will be interesting to com- pare these values with experimental data as and when they become available. The value of enthalpy, H, obtained from MD simulation for α-quartz (−5613 kJ/mol) is in good agreement with those obtained by Murashov and Svishchev who report a value of about −5620 kJ/mol for MD simulations carried out with the BKS potential.72 Note that the lowest enthalpy is obtained for α-quartz which is the most stable phase as indicated by our variable shape NPT ensemble simulations. The enthalpy values for glass (molar volume = 27.27 cm3 /mol) are from experimental data by Piccione et al.73 In the same paper, they also report the enthalpy for crys- talline tridymite (molar volume = 27.52 cm3 /mol) to be −910.00 kJ/mol. This is about 8.2 kJ/mol lower than the glassy silica with similar density. We also find that the en- thalpies of our simulated glasses are higher than that of the corresponding crystalline polymorphs at the same density. Glass-A and Glass-T have enthalpies of 49.55 kJ/mol and 45.14 kJ/mol higher than that of α-quartz and tridymite, re- spectively. 0 10 20 30 40 50 60 ΔH q 298.15 ,ΔG q 298.15 (kJ/mol) Enthalpy Free energy 10 20 30 40 50 V (cm 3 /mol) -3 -2 -1 0 1 2 3 TΔS q 298.15 (kJ/mol) Entropy 1 2 3 4 5 6 8 12 13 1 2 3 4 5 6 7 8 9 10 11 12 13 1 2 3. 4 5 6 8. 11 12 13 g g g 7 9 10 11 7 9 FIG. 9. Plot of Hq 298.15, Gq 298.15, and T Sq 298.15 vs. molar volume for different silica polymorphs using experimental data available in the literature. Here the Hq 298.15, Gq 298.15, and T Sq 298.15 are the formation enthalpy, free energy, and entropy of the given polymorph minus the same quantity for α-quartz. Data for the following polymorphs are shown in the figure: (1) stishovite, (2) coesite, (3) α-quartz, (4) cristobalite, (5) tridymite, (6) ZSM- 23, (7) ZSM-12, (8) ZSM-5, (9) ZSM-11, (10) SSZ-24, (11) zeolite beta, (12) faujasite, and (13) EMT. The data corresponding to glass are marked with g. Metastability is associated with many interesting phe- nomena such as polymorphism and superconductivity.74 The metastability of most polymorphic forms of silica arises from rather small differences in free energy. We have compiled the entropic contribution as well as the free energies and en- thalpies of formation from experimental data available in the literature.34,35,73,75 The relative values of these quantities with respect to α-quartz are plotted as a function of molar volume in Figure 9 for a number of polymorphs for which data are available. For the polymorphs shown in the figure, we note that the entropic contribution to free energy is less than 10%. While T S◦ 298.15 varies only over a range of about 6 kJ/mol, H◦ 298.15 varies over a range of almost 60 kJ/mol (see Figure 9). That is, the entropic contribution to metastability is very small. Thus, although we have not computed the free energies of the various silica polymorphs, it is likely that the trend in free energy will be dictated largely by the trend in the enthalpy. Hence, enthalpy as well as the total interaction energy we have obtained here from MD are indicative of the mestability of the various silica polymorphs. It is satisfying to note that the results from simulation predict the order of metastability of the various polymorphs of silica correctly, in agreement with experiment. Further, these studies show that there are two different regimes of enthalpy dependence on molar volume, giving rise to two distinct types of metastabil- ity: the one at high pressures is dictated by the short-range in- teractions and the one at lower pressures by both short-range and long-range interactions. V. CONCLUSIONS This simulation study provides a comprehensive inves- tigation of 13 different silica polymorphs, with a focus on This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Fri, 04 Sep 2015 07:33:02
  11. 11. 244512-10 Rajappa et al. J. Chem. Phys. 140, 244512 (2014) high-density phases. There are several interesting findings. First, the present study indicates that the BKS potential can predict the lattice parameters of all the 13 phases of silica in agreement with reported X-ray diffraction values. The modi- fied BKS was crucial in reproducing the high density phases. Further, the predicted values of the thermodynamic enthalpies of low density phases are in good agreement with experimen- tal data. This study predicts the enthalpies of the high-density polymorphs for the first time—these can be verified as and when experimental thermodynamic data become available. The predicted enthalpies of CaCl2-type, α-PbO2-type, and pyrite-like phase of silica are, respectively, 4142.03, 4132.19, and 2597.64 kJ/mol. Note that the enthalpies of CaCl2-type, α-PbO2-type are rather close to each other. This is consistent with the fact their molar volumes are not very different. Re- cently, there has been growing interest in prediction of high- pressure crystalline phases.76,77 Our study shows that amorphous phases of silica are as- sociated with lower short-range interaction energies than their crystalline counterparts at the same density. This interesting behaviour suggests a possible reason why silica glasses are so commonly seen. The crystalline phase, however, exhibits long-range order (unlike the glassy phase) and is, therefore, associated with a predominantly favourable long-range ener- getic contribution. For high-density polymorphs, the enthalpy is seen to ex- hibit a marked change from a weak to a steep dependence on molar volume—experimental data indicate that the enthalpy of stishovite is significantly higher than that of coesite. The present study reports enthalpy for polymorphs denser than stishovite and we show that the reason for the sharp change in dependence of the enthalpy on molar volume at higher densi- ties has its origins in the larger contributions from short-range interactions. We further demonstrate that the increased short- range contribution to enthalpy at high densities arises from (i) an increase in coordination number of silicon atom from four to six, as well as (ii) reduced Si–O bond lengths. Since most of these are unfavourable interactions, this leads to the observed steep increase in energy with decreasing molar volume. ACKNOWLEDGMENTS J.G. wishes to thank INSA, New Delhi, for the award of a Senior Scientist position. C.R. thanks DST, New Delhi, for the award of a grant (DSTO1225) under the Women Scientist Scheme. S.Y. thanks DST, New Delhi, for a Raja Ramanna fellowship (DST1065) and grant under nano-mission for 100 teraflop computing facility (DST1169). 1S. Tsuneyuki, M. Tsukada, and H. 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