Comparison of model potentials for molecular dynamics simulation of crystalline silica [Elektronische Ressource] / vorgelegt von Daniel Herzbach

Daniel Herzbach

Sujets

• Physik

Informations

• Publié par: johannes_gutenberg-universitat_mainz
• Publié le: 01 janvier 2004
• Nombre de lectures: 12
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

Comparison of model potentials for molecular
dynamics simulation of crystalline silica
Dissertation
zur Erlangung des Grades
,,Doktor der Naturwissenschaften“
am Fachbereich Physik
der Johannes Gutenberg–Universitat¨
Mainz
vorgelegt von
Daniel Herzbach
geboren inFrankfurt(Main)
Mainz, im Juni 2004Datum der mundlichen¨ Prufung:¨ 05.07.2004Abstract
Simulating SiO with the two body potential developed by van Beest, Kramers, and van Santen2
(BKS) produces many satisfactory results but also characteristic flaws. We investigate these
failures of the BKS potential and compare the performance of the BKS potential with that of
two recently suggested potential energy surfaces that effectively incorporate many body inter-
actions. One approach, which is called the fluctuating charge model, allows the ionic charges
to adjust depending on the chemical environment. The other approach assumes fixed, effective
charges but allows for inducible dipole moments on the oxygen atoms.
The emphasis in this work is placed on situations where BKS fails. We show that an anomaly
in the ratio of quartz’s two independent lattice constantsa andc, which is observed experimen
tally at the transition between and quartz, is missing with BKS. Cristobalite and tridymite
appear mechanically unstable with BKS when periodic boundary conditions are employed
that are compatible with competing high density silica polymorphs. Lastly, the BKS density
of states (DOS) shows characteristic discrepancies from the true DOS.
The fluctuating charge model slightly improves the phononic density of states and correctly
produces stable cristobalite, but it does fail to show the experimentally observedc/a anomaly
at the- transition. Moreover, many properties are reproduced much less satisfactorily than
with BKS.
The fluctuating dipole model remedies all mentioned artifacts. We confirm the view that the
proper behaviour in thec/a ratio is due to the distortion of SiO tetrahedra. These distortions4
can in turn be shown to be due to the many body effects incorporated in the fluctuating dipole
potential.
In addition, the pressure induced phase transition in quartz is studied. All three models show
a transition at similar pressures to the same crystalline phase, which is probably the same as
that found experimentally for the high pressure polymorph called quartz II. On decompres
sion, the BKS potential predicts the formation of an unknown phase at ambient pressures,
while in the two other approaches the quartz II phase reverts to quartz with an intermediate
phase similar to quartz II.
Furthermore we show that the fluctuating charge potential is the only known model potential
to predict the right pressure for a phase transition between two sixfold coordinated stishovite
polymorphs.
We suggest two different methods to calculate piezoelectric coefficients in classical molecular
dynamics simulation in the constant stress ensemble. We find that BKS reproduces experi
mental data reasonably well. However, the results for the fluctuating dipole potential turn out
to underestimate the experimental data by more than 50%, unless the coupling of the electrical
field to the dipoles is switched off. We also show that there is a strong correlation between the
magnitude of the dipoles and the bond bending angles on the oxygen atoms and speculate that
quantum chemical effects that are non electrostatic in nature were (successfully) parametrized
as inducible dipoles. With this interpretation, the fluctuating dipole potential always appears
to be closest to available experimental data out of the three approaches.
iContents
Introduction 1
1. Simulation Techniques and Model Potentials 6
1.1. Molecular Dynamics Techniques . . . . . . . . . . . . . . . . . . . . 6
1.1.1. Parrinello Rahman Barostat . . . . . . . . . . . . . . . . . . 7
1.1.2. Gear Predictor Corrector Integrator . . . . . . . . . . . . . . 8
1.1.3. Langevin Thermostat . . . . . . . . . . . . . . . . . . . . . . 9
1.1.4. Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.5. Ewald Summation . . . . . . . . . . . . . . . . . . . . . . . 10
1.2. BKS Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3. Fluctuating Charge Potential . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1. Charge Equilibration Algorithm . . . . . . . . . . . . . . . . 12
1.3.2. Direct solution . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.3. Extended Lagrangian . . . . . . . . . . . . . . . . . . . . . . 15
1.3.4. Morse Stretch QEq Potential . . . . . . . . . . . . . . . . . . 16
1.3.5. Alternative Morse Stretch fluc Q Potential . . . . . . . . . . 18
1.3.6. Kinetic Parameters for Extended Lagrangian . . . . . . . . . 19
1.4. Fluctuating Dipole Model . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5. Comparison of Performance . . . . . . . . . . . . . . . . . . . . . . 22
2. The- Quartz Transition 24
2.1. Transition Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2. Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3. Thermal Expansion and c/a Ratio . . . . . . . . . . . . . . . . . . . . 30
2.3.1. Influence of Specific Properties of the Potentials . . . . . . . 32
2.4. Distortion of Tetrahedra . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.1. Influence of Atomic Bond Character . . . . . . . . . . . . . . 37
2.5. Elastic Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6. Phonon Density of States . . . . . . . . . . . . . . . . . . . . . . . . 43
3. Pressure Driven Transitions in Quartz 47
3.1. Quartz II Transition in Simulations . . . . . . . . . . . . . . . . . . . 48
3.1.1. T Path Ways . . . . . . . . . . . . . . . . . . . . . . 49
3.2. Quartz II b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3. X Ray Diffraction Spectra . . . . . . . . . . . . . . . . . . . . . . . 52
iiiContents
3.4. Other High Pressure Phases . . . . . . . . . . . . . . . . . . . . . . 54
4. Silica Polymorphs Other Than Quartz 56
4.1. Cristobalite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2. Tridymite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3. Stishovite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.1. High Pressure Behaviour . . . . . . . . . . . . . . . . . . . . 61
5. Electromechanical and Dielectric Properties 63
5.1. Theory and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1.1. Linear Response . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1.2. Ambiguity of the dipole and its fluctuation . . . . . . . . . . 67
5.1.3. Fluctuation estimators for dipoles . . . . . . . . . . . . . . . 68
5.1.4. Direct estimators with noise reduction . . . . . . . . . . . . . 69
5.2. Temperature Dependence of d in Quartz . . . . . . . . . . . . . . . 7111
5.3. Pressure of d in Quartz . . . . . . . . . . . . . . . . 7411
5.4. Bond Angle of the Dipole Moment . . . . . . . . . . . . 75
6. Conclusion and Outlook 77
A. Calculations 81
A.1. Calculations for Fluc Q Potential . . . . . . . . . . . . . . . . . . . . 81
A.1.1. Direct Solution of Charge Equilibration Equations . . . . . . 81
A.1.2. Two Body Slater Integral . . . . . . . . . . . . . . . . . . . . 82
B. Programs 85
B.1. Molecular Dynamics Code . . . . . . . . . . . . . . . . . . . . . . . 85
B.1.1. Inputfiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
B.2. Analysis Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
List of Figures 158
Bibliography 164
ivIntroduction
Almost a century ago, the discovery of quantum mechanics allowed one to describe the
fundamental interactions of ions and electrons in matter. Even some decades earlier,
the equations of statistical mechanics were formulated, which relate the microscopic
interactions of ions and electrons to the resulting macroscopic properties of a given
material. However it is generally not possible to solve these equations analytically for
complex materials in order to predict their properties. Dirac expressed the problem
in 1929 as follows: “The fundamental laws necessary for the mathematical treatment
of large parts of physics and the whole of chemistry are thus fully known, and the
difficulty lies only in the fact that application of these laws leads to equations that are
too complex to be solved.” This statement still holds despite the increasing power of
computers. Simplified models for the interactions need to be formulated for analytical
theories. More approximations are required if one wants to predict the behaviour of
materials beyond the harmonic approximation. If theory and experiment disagree, it is
often difficult to say at what point theory failed.
The use of computers made it possible to reduce this uncertainty, as computer simu
lations made it possible to solve the equations for microscopic empirical model po
tentials with (theoretically) arbitrary accuracy, thus allowing us to see how the macro
scopic system behaves as a whole without the need to rely on further assumptions.
The numerical computation of macroscopic observables can be realized by sampling
randomly over phase space in a Monte Carlo (MC) simulation [58] or by following
the trajectory of Newton’s equations of motion, and thus sampling over phase space as
well, in