Investigation of Defect-Assisted Material

INVESTIGATION OF DEFECT-ASSISTED

MATERIAL TRANSPORT IN MAGNESIUM

OXIDE BY MOLECULAR SIMULATIONS

ADRIAAN ANTHONY RIET

Submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

Chemical Engineering

CASE WESTERN RESERVE UNIVERSITY

August 2020

CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the thesis of

ADRIAAN ANTHONY RIET

Candidate for the degree of Doctor of Philosophy

Committee Chair

Daniel J. Lacks, PhD

Committee Member

James A Van Orman, PhD

Committee Member

Burcu Gurkan, PhD

Committee Member

Chris L Wirth, PhD

Date of Defense

June 12, 2020

*We also certify that written approval has been obtained for any proprietary material

contained therein. This work is dedicated to my wife Heather, without whom I could never have done this. Table of Contents

List of Tables ...... 7

List of Figures ...... 8

Acknowledgements ...... 12

Abstract ...... 1

1 Introduction ...... 3

1.1 Why study crystalline defects ...... 3

1.2 What is classical molecular dynamics simulation ...... 4

1.3 Brief overview ...... 9

2 Molecular Dynamics Simulation of Vacancy-Solute Binding Free Energy in

Periclase ...... 11

2.1 Chapter Summary ...... 11

2.2 Introduction ...... 11

2.3 Methods ...... 13

2.4 Results ...... 15

2.5 Discussion ...... 20

2.5.1 Energy of binding ...... 20

2.5.2 Diffusion ...... 24

3 Structure Diffusivity Relations at Grain Boundaries in Polycrystalline MgO ...... 25

3.1 Chapter Summary ...... 25 3.2 Introduction ...... 26

3.3 Methods ...... 30

3.4 Results ...... 34

3.5 Discussion ...... 41

3.5.1 Estimate of diffusion coefficients ...... 41

3.5.2 Elucidation of elementary steps in the diffusion process at grain boundaries

3.5.3 Possible non-Arrhenius diffusive behavior at grain boundaries ...... 47

4 A Molecular Dynamics Study of Grain Boundary Diffusion in MgO ...... 49

4.1 Chapter Summary ...... 49

4.2 Introduction ...... 51

4.3 Methods ...... 53

4.3.1 The polycrystalline system ...... 53

4.3.2 Interatomic forces ...... 57

4.3.3 Simulation methodology ...... 58

4.3.4 Property analysis ...... 59

4.4 Results ...... 62

4.4.1 Structure ...... 62

4.4.2 Diffusion ...... 67

4.5 Discussion ...... 72

4.5.1 Comparison with experimental data ...... 72

4.5.2 Temperature dependence of grain boundary diffusivity ...... 74

4.5.3 Pressure dependence of grain boundary diffusivity ...... 76

4.6 Conclusions ...... 80

5 An analysis of dominant transport mechanism as a function of grain size ...... 82

6 Appendix ...... 85

6.1 Algorithms/Scripts used (python code) ...... 85

6.1.1 Identification of vacant site ...... 85

6.1.2 Grain boundary lattice creation ...... 87

6.1.3 Grain boundary width determination ...... 90

6.1.4 Grain boundary atom assignment and grain assignment ...... 92

7 Works-Cited List ...... 96

List of Tables

Table 2.1: Free energy of binding for the vacancy and defect cation as a function of temperature and pressure. Parenthesis denote 2σ uncertainties in the last significant digit(s) ...... 19

Table 4.1: Overall diffusion coefficients, grain boundary widths, and grain boundary fractions from simulations. Diffusivity is overall diffusivity as obtained by a fit to the

Einstein diffusion equation. The values in parenthesis are 1σ standard errors in the last significant figure...... 67

Table 4.2: Activation volumes as a function of temperature and pressure. Uncertainties are 1σ standard errors...... 78

Table 4.3: Parameters for second order fit of activation volume at 2000 C for a second order fit of the activation volume. Ranges indicate 95% confidence intervals. Other uncertainties are 1σ standard errors...... 78 List of Figures

Figure 2.1: Radial distribution function for vacancy-defect cation interaction as a function of A) temperature at 0 GPa, and B) pressure at 2400 K. Uncertainties are two-sigma standard errors...... 16

Figure 2.2: Potential of mean force for the vacancy-impurity cation interactions as a function of A) temperature at 0 GPa, and B) pressure at 2400 K. Uncertainties are two- sigma standard errors...... 17

Figure 2.3: Vacancy-solute separation distances. The red sphere is the cation defect, and open spheres are the vacant sites. Black spheres are magnesium atoms, and teal spheres are oxygen atoms. A) Nearest-neighbor separation, corresponds to the energy minimum plotted in Figures 1 and 2...... 18

Figure 2.4: Binding energy for vacancy-defect cation interactions as a function of A) temperature, and B) pressure. Error bars are two-sigma standard errors, and highlighted regions indicate the 95% confidence interval of the fit at the conditions specified ...... 20

Figure 2.5: Binding energy along a mantle adiabat, assuming a 0.4 K/km temperature gradient and PREM. Estimated binding energies obtained by a fit to the data in Table 2.1.

Shaded region denotes 95% confidence interval. Dotted lines are projected binding energies given different binding enthalpies, along the mantle adiabat...... 23

Figure 3.1: Depiction of a typical grain boundary. A) Model/equilibrium structure, representative of the lattice at 0 K. B) Thermalized structure. C) Thermalized structure imposed on equilibrium structure...... 27

Figure 3.2: Initial setup of the simulated system. Four distinct grain boundaries exist. The tilt axis is in the y direction, with a grain boundary angle of 42°. The z direction is normal to the grain boundary plane. Large spheres are oxygen ions, while small spheres are magnesium ions. Atoms are colored by their distance to their 6th nearest neighbor (blue atoms have their sixth nearest neighbor at ~2.1 Å, red atoms have their sixth-nearest neighbor at ~3.5 Å.) The grain boundary angle between two adjacent grains, defined as the angle needed to rotate the lattice of one grain onto the other, is ~42°. The resulting grain boundary is a symmetric tilt boundary of 24° around [100]...... 31

Figure 3.3:Probability distribution for the six nearest Mg-O distances around an Mg atom. (a) results for configurations during the molecular dynamics trajectory at 1800 °C.

(b) results for inherent structures obtained from the molecular dynamics trajectory at

1800 °C...... 35

Figure 3.4 (a) Configuration of an inherent structure, where atoms are colored by the distance to their sixth-nearest neighbor. (b) Spatial distribution of undercoordinated atoms in an inherent structure from the simulation at 1800 °C. (c) Close up view of the results in part b, near one grain boundary...... 36

Figure 3.5: Fraction of the Mg atoms in the system that are undercoordinated. The curves are results for 1300 °C (blue, bottom), 1500 °C (orange, middle), 1800 °C (gray, top). .. 38

Figure 3.6: (a-c) Fraction of undercoordinated atoms in an inherent structure that were also in the inherent structure that the system was in 1 ns earlier. (d-f) Fraction of undercoordinated atoms in an inherent structure that were also in the inherent structure at time t=0. In all cases the blue curve represents magnesium and the orange curve represents oxygen...... 39

Figure 3.7: One-dimensional mean-squared displacements as a function of time. Blue data is for the x-dimension (in the grain boundary plane perpendicular to the tilt axis),

orange data is for the y-dimension (down the tilt axis), and gray data is for the z- dimension (perpendicular to the grain boundary plane). A dashed line with a slope of unity is given as a reference...... 40

Figure 3.8: Mean squared displacements as a function of time, for Mg atoms at 1800 °C.

Results are shown for all Mg atoms, and for Mg atoms that spent the entire simulation only in the bulk crystal (i.e., not at a grain boundary)...... 43

Figure 3.9: Example of a transition between two inherent structures during a simulation

(color online). Frames (a) and (b) show atomic positions (large spheres are oxygen; small spheres are magnesium) 1 ps apart. The major change in atomic positions during the transition involves an oxygen atom (labeled 1) jumping to fill a nearby vacancy.

However, there is an associated shift of an adjacent oxygen atom (2) to a different, lower coordination position within the grain boundary, and a network of smaller motions of other nearby atoms within the grain boundary as well, extending beyond the range of the image ...... 45

Figure 3.10: Mean potential energy of the inherent structures as a function of temperature. Error bars denote standard error, which is determined by sampling four independent time intervals per simulation. Changes in the mean potential energy of the inherent structures have been related to super-Arrhenius changes in dynamical properties, and thus these results suggest that the diffusion constant at grain boundaries may similarly follow a super-Arrhenius temperature dependence...... 48

Figure 4.1: Representative equilibrated structure. View is normal to the angle of rotation of the grain boundaries. Red spheres are oxygen ions, and white spheres are magnesium ions...... 54

Figure 4.2: Distributions of distances from a central ion the 6th closest ion. The first peak is cut off in order to more clearly see the second peak. In part a), thermal denotes the non- energy minimized distribution...... 64

Figure 4.3: Fraction of undercoordinated atoms as a function of (a) temperature, (b) pressure...... 65

Figure 4.4: Potential energy of the energy minima visited by the polycrystalline system, at P = 0 GPa. In comparison, the energy of the single crystal is shown (which increases with temperature due to thermal expansion)...... 66

Figure 4.5: Density near the grain boundary at three different pressures. The results at

P=0, 10 and 50 GPa are at temperatures of 1800, 2300 and 2000 °C respectively...... 67

Figure 4.6:Diffusion coefficients as a function of temperature. Closed symbols are present results for grain boundary diffusion, and open symbols are previous simulation results for diffusion in MgO melts (Lacks et al., 2007). Error bars indicate standard error

...... 71

Figure 4.7: Grain boundary diffusion coefficients as a function of pressure. Error bars represent the standard error of the estimate of the diffusion coefficient...... 79

Figure 5.1: grain sizes where lattice diffusion and grain boundary diffusion contribute equally to transport given a hypothetical defect cation with 100 ppm concentration...... 84

Acknowledgements

This material is based upon work supported by the National Science Foundation under grant number EAR-1250331. The calculations were carried out using computational resources through the Ohio Supercomputing Center and the High Performance

Computing Resource in the Core Facility for Advanced Research Computing at Case

Western Reserve University. I would like to thank Jonathan Gillispie and Branden Kraus for working on this project with me. Of course, I owe the greatest debt to my advisors and mentors in this project, Jim Van Orman and Dan Lacks.

Investigation of Defect-Assisted Material

Transport in Magnesium Oxide by

Molecular Simulations

Abstract

ADRIAAN ANTHONY RIET

Solute-vacancy interactions and grain boundary structure and dynamics in an MgO crystal are investigated through molecular dynamics simulations. For the first time using molecular dynamics simulations, the binding entropy and enthalpy are determined directly for a solute-vacancy system in a single crystal of magnesium oxide, with a binding entropy of 13±5 (95% CI) J/mol K. The binding energy is also shown as a function of pressure. The method of (Sastry, Debenedetti, Stillinger, et al.) to quantify structure in glasses is applied to simulations of MgO grain boundary structures to identify equilibrium grain boundary structure and grain composition. The dynamical exchange of atoms within the grain boundary is demonstrated. The grain boundary diffusion coefficient is obtained as a function of temperature and pressure, and implications for grain boundary diffusion and transport through the inner earth are presented, with the

1 result that the characteristic grain boundary diffusion length is constrained to be less than about 100 m for magnesium and oxygen at the core-mantle boundary. Finally, the transition between effective volume diffusion and effective grain boundary diffusion is obtained as a function of temperature and pressure.

1 Introduction

1.1 Why study crystalline defects

Crystalline and glassy materials are of special importance as barrier materials as they are stable at high temperatures and pressures, their chemical bonds are not easily broken, and material transport is generally slow within them. Thin films of non-diffusive material are also of critical importance for use in semiconductors, as a diffusion barrier prevents the metal on the traces from leaching into the surrounding material, reducing the conductivity of the traces and increasing the conductivity of the barrier material.

Magnesium oxide in particular has been used as a barrier film to shield the cathode of plasma displays from very high energy particles (Kim et al.), and is one approved barrier for long term storage of nuclear waste (Department of Energy). In addition, most of the earth’s composition is solid. Given the size of the earth, and that measurements can only directly be taken on the surface, obtaining information about the internal structure of the earth requires accurate models of its constituent materials.

Constraints on diffusivity through the various modes are important, especially as there is feedback between the modes of transport. For example, the amount of chemical transport in the inner earth is dependent on mass transport through creep, volume and grain boundary diffusion. The total flux through creep and grain boundary diffusion are dependent on grain boundary diffusivity and the grain size, but the grain size is also limited by grain coarsening, which is a function of grain boundary diffusivity and creep, and all of these may be functions of impurity concentration. Because of this, if we have an accurate model of grain boundary transport for geological materials, we can constrain

3 chemical transport and grain size within the solid earth. Alternately, if there is knowledge of grain size and composition in a layer of the earth that is obtained using an orthogonal method, volume and grain boundary diffusivity can be used to constrain the history of the layer.

This work addresses defect-assisted diffusion in magnesium oxide, MgO.

Although commonly used as a model material for oxides because of its simple crystal structure and phase stability, MgO is in important material in its own right. Geologically speaking, MgO is the mineral periclase, and is thought to be a minor component in the earth’s inner mantle. It may be of especial importance in thin, ultra-low seismic velocity zones which lie in places along the earth’s core-mantle boundary (Garnero et al.; Wicks et al.). MgO is especially useful as a model material for high pressure applications as no phase transitions are observed until over 400 GPa (Karki et al.), which is above the pressure of the core-mantle boundary.

1.2 What is classical molecular dynamics simulation

“All models are wrong. Some are useful” – George Box (attributed).

It is generally understood that the physical world in which we live follows the laws of relativistic quantum mechanics. This implies that we can completely mathematically describe any system by applying Schrodinger’s equation to the system with the correct boundary conditions. A critical drawback of this approach is that analytic solutions to Schrodinger’s equation are only possible for simple toy problems and very simple systems like a single hydrogen atom. With certain approximations, however, we can simulate systems involving up to a few hundred atoms and their interactions over timescales of up to a few nanoseconds by numerically approximating the solution to the

4 wave function. This is done iteratively, by using an instantaneous solution to identify the forces on each atom, repositioning the atoms slightly, and recalculating the forces. In this manner we can integrate through time, an approach called first principles molecular dynamics. This approach is limited because solving the wave function of an electron requires knowledge of the entire system, resulting in a scaling factor of O(n2ln n) or O(n3) depending on the system, where n is the number of electrons.

If we can omit electronic interactions, or in other words, if the effective force between atoms is solely a function of distance between atom centers, we can treat atoms as particles with effective interaction energies that are only a function of inter-particle distance and use Newton’s equations of motion to describe the system. This becomes essential to model larger systems and longer timescales, which are necessary to study emergent properties of bulk materials and dynamic interfaces. In this way, we use a model that is technically “wrong”, but ultimately very useful, though these models must be validated to ensure accuracy in the results. This approach is called classical molecular dynamics.

There is a great deal of flexibility allowed by classical mechanics, as interactions of any kind between particles can be modelled, provided the attractive and repulsive forces on the particles are correctly specified. Modelling of these forces and the potential energy landscapes they imply, hereafter referred to as “potentials”, is typically done by fitting to the results of ab initio quantum mechanical calculations and/or by comparing the properties exhibited by the resulting system to experiments. Proper design of these interactions forms the association between the hypothetical particles in the simulation and real-world materials.

The interaction model used in this work is obtained partially from both methods:

An initial oxygen-oxygen interatomic potential was obtained by ab initio (quantum calculations) data for oxygen in silicates and aluminophosphates (van Beest et al.); and the magnesium-oxygen interatomic potential was obtained by fitting to experimental data such that the transport properties of the resulting material corresponded with experimental observations (Daniel J Lacks, Rear, et al.; Speziale et al.). Perhaps interestingly, the magnesium-magnesium interaction is well described by a simple coulombic model, so additional attractive/repulsive forces are not modeled here, as with other potentials used for MgO where interatomic potentials were specified by species- species interactions (Sun et al.; Gourdin and Kingery).

Short-range potentials (including Coulombic interactions) are often calculated directly in real space, however some transforms can increase computational efficiency.

For example, long range coulombic interactions can be calculated more efficiently in

Fourier space (but short range interactions become intractable). A cutoff distance is thus chosen to minimize the computational load, such that the short-range interactions are computed in real space and interactions over distances longer than the cutoff are computed in Fourier space. This method is computationally robust provided that there is no excess charge in the system (Darden et al.). Coulombic potentials are often computed this way, and that is the approach taken in this work.

Once the interaction potentials between particles have been determined, an initial configuration of the system must be specified. Often this initial configuration is highly ordered or energetic, and the forces on the particles must “play out” for some period of time while the system equilibrates to configurations representative of the state in phase

6 space (the collection of all possible positions and momenta for all particles within the system under the given thermodynamic conditions). Following the principles of statistical mechanics, the phase space a system explores is dependent on the thermodynamic properties that are conserved.

For example, the simplest case is where the energy, volume and number of particles in a system are conserved. This is called the microcanonical ensemble, and is the result of applying Newton’s equations of motion to a system of particles. In equilibrium, this system will sample the collection of particle positions and momenta that maximize the entropy of the system.

If, instead of conserving energy, we allow our particles to exchange energy with an outside group of particles held at a constant temperature, we obtain the canonical ensemble. The canonical ensemble, in equilibrium, will sample the collection of particle positions and momenta that minimizes the free energy of the system, which is the amount of useful work obtainable at constant volume and temperature. Other properties such as the total chemical potential or the enthalpy can be conserved if other ensembles are desired, which will change the analysis.

With the correct analysis and infinite precision, any ensemble can be used to calculate any thermodynamic property, as the state of the system is fully described.

However, it is generally appropriate to use an ensemble that matches conditions usually seen in experiments (e.g. constant pressure or volume and temperature), to aid in the usefulness of extracted information.

With the right interaction between particles and the ensemble appropriate for a system chosen, the boundary conditions must be addressed. Without some form of

7 boundary conditions, all simulations would effectively take place in a vacuum, and while the earth does exist in a vacuum, merely counting each particle required for a simulation of the earth’s size would take a modern computer longer than the projected life of the earth. The possibilities for boundary conditions are as flexible as the possibilities for interaction potentials between particles, but a common practice is to set periodic boundary conditions.

With periodic boundary conditions, when a particle moves past a given cutoff coordinate, called the cell dimension, the numerical value of its position is decreased by the cell dimension, and similarly particles that move into space below a zero coordinate have their position increased by the cell dimension. Particles near the cutoff coordinate interact with particles near the opposite cutoff as if they were adjacent. This effectively tessellates the simulation cell to create an infinitely large system. Care must be taken to ensure that a particle does not interact with a periodic image of itself, as this would lead to artifacts in the simulation results.

With an initial configuration, the right boundary conditions, and an appropriate interaction potential, molecular statics calculations are possible. To incorporate an appropriate ensemble and use molecular dynamics, a time integration method must be used. This is done by repeatedly calculating the static forces on the particles, and then moving the particles in the direction of those forces as if some time had elapsed. To avoid physically unrealistic results, the elapsed time between recalculations must be smaller than the vibrational frequency of the lightest atom in the simulation. Otherwise, one of the light atoms would eventually travel too close to the core of another atom, a large force would be calculated, and if the simulation continued, the atom would impart significant

8 energy to the rest of the atoms within the simulation, a process commonly referred to as

“blowing up.” In practice, the effective time between force calculation steps is on the order of femtoseconds, which limits the timescale of simulations as only up to ~1015 calculations may be done in a reasonable time on modern computer systems. The method of time integration is a very important part of a molecular dynamics simulation, and it is strongly recommended to use algorithms that have been peer reviewed and scale well.

1.3 Brief overview

In Chapter 2, we find good agreement between the model for MgO we use in our molecular dynamics simulation and previous experimental and theoretical work, which adds confidence to both our model and the experimental results. We also determine, for the first time, the entropy and enthalpy of binding for a vacancy-solute defect pair through simulation.

At lower mantle conditions, grain boundary diffusion has been proposed as a possible mechanism for interaction between the earth’s core and mantle (Hayden and

Watson, “A Diffusion Mechanism for Core–Mantle Interaction”; Hayden and Watson,

“Grain Boundary Mobility of Carbon in Earth’s Mantle: A Possible Carbon Flux from the

Core”), with a characteristic diffusion length on the order of 1-100 km. Grain boundary diffusion is a difficult phenomenon to model, as it is a mathematically stiff problem involving vastly different timescales. Because of the extreme conditions of the earth’s mantle, it is difficult to measure grain boundary diffusion at those conditions experimentally.

In Chapter 3, we demonstrate a model for measuring grain boundary diffusion directly and atomistically in a molecular dynamics system, and validate experimental

9 estimates of the thermodynamic properties of diffusants in magnesium oxide. Then in

Chapter 4 we apply this model to temperatures and pressures relevant to the earth’s mantle, by directly modeling grain boundary diffusion in MgO by classical molecular dynamics simulation, and use this to constrain the possible diffusive exchange in the lower mantle. The domains where grain boundary diffusion is important are dependent on the relative magnitude of the bulk crystalline and grain boundary diffusion coefficients.

Taken together, the results of these molecular dynamics simulations allow estimation of where intracrystalline diffusion transports more or less material than grain boundary diffusion as a function of grain size, which is the subject of Chapter 5.

2 Molecular Dynamics Simulation of Vacancy-Solute Binding Free

Energy in Periclase

2.1 Chapter Summary

The free energy of binding between a cation vacancy and charge-balancing tetravalent cation in MgO is calculated from molecular dynamics simulations, at temperatures of 1200-2400 K and pressures 0-25 GPa. The entropy of binding, obtained from the temperature dependence of the binding free energy, is found to be independent of pressure and to have a value of 13(5) J/mol K (2σ). The binding volume, obtained from the pressure dependence of the binding energy, is independent of temperature and has a value of 1.7(3) cm3/mol (2σ). These results are in excellent agreement with experimental estimates for several different trivalent cations, showing that the binding entropy and volume are not strongly dependent on the identity or charge of the cation.

The binding energy is predicted to increase with depth along a mantle adiabat, leading to lower concentrations of unbound vacancies, and correspondingly slower diffusion rates for unbound cations.

2.2 Introduction

In periclase (MgO), solutes with valence greater than two are accompanied by vacancies on cation sites that are created to balance the excess charge of the solute.

Because the solute and vacancy have opposite effective charges, they experience a coulombic attraction that tends to bind them as pairs on neighboring cation sites

(Sempolinski and Kingery; Carroll et al.; James A. Van Orman et al.; Crispin and Van

Orman). Similar solute-vacancy binding occurs in many other crystalline materials, including halides (Bassani and Fumi), olivine (Jollands et al.) and metals (Yamamoto and

Doyama; Klemradt et al.; Hoshino et al.; Ohnuma et al.).

The formation of solute-vacancy pairs has strong effects on diffusion and other properties that involve vacancy hopping. It enhances the diffusivity of the solute by strongly increasing the availability of a vacant site to hop to, while inhibiting the mobility of the vacancy and of other cations that utilize vacancies for diffusion. The reorientation of solute-vacancy pairs in a stress or electric field also contributes to anelastic and dielectric relaxation, respectively (Dreyfus and Laibowitz).

Previous work has attempted to quantify the interaction between vacant sites and impurities in MgO. Sempolinski and Kingery estimated the binding energy between Sc3+ and cation vacancies from electrical conductivity measurements in Sc-doped MgO samples over a range of temperatures (Sempolinski and Kingery). A second approach involves solute diffusion experiments in which the free energy of binding is inferred from the concentration dependence of the diffusion coefficient. The binding energy is obtained by fitting the experimental concentration profile based on a five-frequency vacancy hopping model and mass-action law (Lidiard; Perkins and Rapp), and has been applied to estimate the vacancy binding energies for Al3+, Ga3+ and Cr3+ in MgO (James A. Van

Orman et al.; Crispin and Van Orman). A similar method has been applied to determine the Cr3+-vacancy binding energy in olivine (Jollands et al.). The experimental data provide some constraints on the temperature and pressure dependence of the binding energy, but with large uncertainties.

Current methods used to estimate the binding energy from an atomistic perspective generally involve ab initio static calculations, with the assumption that the enthalpy of binding at some temperature is equal to the internal energy difference between the bound state and the free state at zero kelvin (De Vita, M J Gillan, et al.; De

Vita, M. J. Gillan, et al.; Hoshino et al.; Ammann et al.). While first principles molecular dynamics simulations theoretically could resolve the enthalpy of binding directly, the computational cost prohibits the investigation for large binding energies (Løvvik et al.).

Here we use classical molecular dynamics to directly determine the free energy of binding between vacancy and impurity cation at temperatures and pressures relevant to experiments. From these free energy results, we can determine the entropy of binding, which governs the temperature dependence, and volume of binding, which governs the pressure dependence. We examine this interaction of the vacancy and impurity cation in the infinitely dilute limit, so as to remove complications due to higher order interactions.

2.3 Methods

The simulation cell is a cubic box that contains 729 (9x9x9) unit cells of MgO periclase. Periodic boundary conditions are used in all dimensions. In order to address the infinitely dilute limit, the system has a single Mg vacancy and a single impurity cation that has a charge twice that of Mg, so only one impurity ion is needed to charge-balance the vacancy. Thus there is a total of 2916 O anions, 2194 Mg cations and one impurity cation.

Atomic forces are calculated using a modified van Beest potential (van Beest et al.; Zhang et al.; Daniel J Lacks, Goel, et al.; Riet et al.). As we are examining generic

13 behavior, the mass and non-Coulombic interactions of the impurity cation are taken to be the same as those for Mg; i.e. the solute has exactly the same properties as Mg but with twice the charge.

Molecular dynamics simulations to study the system dynamics are carried out at constant volume and temperature, after first running a shorter simulation at constant pressure and temperature to determine the equilibrium volume. Temperature is controlled using the Nosé-Hoover thermostat (Nosé; Hoover). The time step for the numerical integration of the equations of motion is 1.5 fs. Simulations are carried out using the

GROMACS software package (H.J.C. Berendsen et al.).

We define the vacancy position in regard to the perfect lattice underlying the crystal. To find the position of the vacancy, the positions of all cations are mapped to the closest site in the underlying lattice; the position of the vacancy is that of the unoccupied lattice site.

We calculate the radial distribution function for the vacancy and the impurity cation, with reference to the position of the defect and the effective position of the vacancy as described previously. The radial distribution function, g(r), represents the relative probability of finding the impurity cation at a distance r away from the site of the vacancy. To get this relative probability we find the time-averaged number density of impurity cations in a spherical shell at a distance r from the vacancy,

〈푛 (푟)〉 휌(푟) = 푑 (2.1) 4휋푟2 ∆푟

where 〈푛푑(푟)〉 is the time averaged number of impurity cations in the shell at distance r and r is the thickness of this shell. The radial distribution is then obtained as

휌(푟) 푔(푟) = (2.2) 휌(푟→∞)

We approximated 휌(푟 → ∞) by an average of 휌(푟) between a cutoff distance rc and half the length of the simulation cell, which is the longest distance for which the calculation is meaningful; we used rc =17 Å, but found that the results are insensitive to the precise value of rc.

The potential of mean force, w(r), can be obtained from the radial distribution function,

푤(푟) = −푘푇 ln 푔(푟) (2.3)

We estimate the uncertainty in the potential of mean force by dividing the simulation into five equal parts, and calculating the potential of mean force for each subset of the simulation. The potential of mean force represents the free energy of interaction between the vacancy and the impurity cation, as a function of separation distance, with infinite separation as the reference state.

2.4 Results

Simulations were run at temperatures between 1200 and 2700 K and pressures between 0 and 25 GPa. First, the equilibrium volumes were determined at each temperature and pressure of interest using constant pressure simulations. At zero pressure, the crystal lattice expands from Mg-O bond length of 2.14 Å at T=1200 K to

2.20 Å at T=2400 K. At T=2400 K, the crystal lattice compresses from 2.20 Å at 0 GPa to

2.06 Å at P=25 GPa. Results for the equilibrium bond length as a function of temperature and pressure are given in Table 2.1.

Molecular dynamics trajectories are then obtained at the equilibrium volumes.

From the trajectories, the radial distribution functions for the vacancy and impurity cation are obtained from Equation 2.1. Results for the radial distribution functions are shown in

Figure 2.1. These results show that the impurity cation is more likely to be found closer to the vacancy, with the highest likelihood of finding the impurity cation at the nearest cationic lattice site to the vacancy. As the temperature increases, the vacancy and impurity cation are not as likely to be close to each other; however, as the pressure increases, the vacancy and impurity cation are more likely to be close to each other.

Figure 2.1: Radial distribution function for vacancy-defect cation interaction as a function of A) temperature at 0 GPa, and B) pressure at 2400 K. Uncertainties are two- sigma standard errors.

The result that the vacancy and impurity cation are more likely to be close to each other than they would be if randomly situated on cation sites implies that there is an effective attractive interaction between them. This effective interaction energy is described by the potential of mean force, obtained from Equation 2.2, and the results are shown in Figure 2.2. The potential of mean force is zero at large separations, becomes increasingly attractive (more negative) as the separation decreases, but then becomes less attractive as the separation gets very small. The minimum potential occurs when the

16 vacancy and impurity cation are at the relative positions shown in Figure 2.3a. Upon closer approach, where the impurity cation straddles two neighboring vacant sites as shown in Figure 2.3b, the potential increases. The value of the potential of mean force at its minimum is the free energy of binding between the vacancy and the impurity.

Figure 2.2: Potential of mean force for the vacancy-impurity cation interactions as a function of A) temperature at 0 GPa, and B) pressure at 2400 K. Uncertainties are two- sigma standard errors.

Results are obtained for w(r) at different temperatures and pressures. The free energies of binding are obtained as the minima on each of these w(r) curves. Since the results are obtained at the equilibrium volume for each temperature and pressure, the binding free energies correspond to Gibbs free energies, and are denoted here as Gbind.

Results for Gbind are given in Table 2.1, and are shown as a function of temperature and pressure in Figure 2.4. The value of Gbind decreases with increasing temperature, and increases with increasing pressure. The temperature dependence of

Gbind is described by the entropy of binding, Δ푆푏푖푛푑,

휕∆퐺푏푖푛푑 ∆푆푏푖푛푑 = − ( ) (2.4) 휕푇 푃

and the pressure dependence of Δ퐺푏푖푛푑 is described by the volume of binding,

Δ푉푏푖푛푑,

휕∆퐺푏푖푛푑 ∆푉푏푖푛푑 = ( ) . (2.5) 휕푃 푇

Table 2.1: Free energy of binding for the vacancy and defect cation as a function of temperature and pressure. Parenthesis denote 2σ uncertainties in the last significant digit(s)

T [K] P [GPa] Mg-O bond length [Å] Gbind [kJ/mol]

1200 0 2.14 -80(12)

1350 0.2 2.15 -80(6)

1500 -0.8 2.16 -80(5)

1650 0 2.16 -76(2)

1800 0 2.17 -74(1)

1800 13.3 2.09 -97(2)

2100 0 2.18 -68(1)

2400 0.1 2.20 -61(2)

2400 16 2.09 -95(3)

2400 24.8 2.06 -107(2)

2700 26.4 2.06 -102(2)

Based on linear regression of Gbind as a function of temperature and pressure, we

3 find that Sbind =13±5(2σ) J/(mol K) and Vbind =1.6±0.3(2σ) cm /mol. This regression was carried out for all data in Table 2.1, as no statistically significant pressure

19 dependence of Sbind or temperature dependence of Vbind was observed (the interaction p-value is 0.44).

2.5 Discussion

2.5.1 Energy of binding

To our knowledge this is the first simulation study to determine the free energy of binding between a vacancy and a charge-balancing impurity ion. The temperature and pressure dependences of this free energy are given by the entropy of binding and the volume of binding, respectively. We now assess the accuracy and relevance of our results by comparison with experiments and previous simulation work.

The magnitude of the binding free energy falls in the range of that found experimentally in MgO periclase. We find binding free energies of 61 – 107 kJ/mol at T

20 between 1200 K and 2400 K and P between 1 atmosphere and 25 GPa. Based on ionic conductivity experiments in Sc-doped samples, there is an experimental estimate a binding energy of 70±20 kJ/mol for vacancy-Sc3+ in the temperature range ~1370-1500 K

(Sempolinski and Kingery). Analysis of Al diffusion profiles in MgO, yielded values between 44 and 96 kJ/mol for the vacancy-Al3+ binding energy at temperatures between

1577-2273 K and pressures between 0-25 GPa (James A. Van Orman et al.). Using a similar approach, values of 58-100 kJ/mol for the vacancy-Ga3+ binding energy in MgO at 1500-2300 K, while experiments at similar conditions yielded much lower vacancy-

Cr3+ binding energy estimates in the range of 15 – 32 kJ/mol (Crispin and Van Orman).

The entropy of binding from our simulations, 13±6(2σ) J/(mol K), also agrees well with experimental results. For Al3+, the binding entropy was constrained to be less than ~25 J/(mol K) (James A. Van Orman et al.). For Ga3+ and Cr3+, entropy of binding was found to be 5.1±22(2σ) J/(mol K) and 7.3±7(2σ) J/(mol K), respectively (Crispin and

Van Orman).

The volume of binding from our simulations, 1.6±0.3(2σ) cm3/mol, falls within the constraints for Al3+ binding of 0-1.8 cm3/mol given by (James A. Van Orman et al.) over the same pressure range (0-25 GPa).

Previous investigations addressed the vacancy-impurity ion binding in MgO with density functional theory (DFT) calculations. This approach differs from ours in that it uses a more rigorous method for calculating the energy, but due to the computational intensity it did not address interactions at finite temperatures. We can compare our results by extrapolating our results to zero temperature – by this extrapolation, we find the enthalpy of binding to be 97(8) kJ/mol, and the lattice parameter at zero temperature to be

4.09 Å. Our enthalpy of binding falls between the DFT enthalpy of binding calculated through DFT calculations for cell sizes of 4.0 Å (73-79 kJ/mol) and 4.2 Å (123-130 kJ/mol) (Ammann et al.).

Therefore, we find that the present results for the free energy of binding of vacancy-impurity ion pairs in MgO periclase are in excellent agreement with experimental results and DFT calculations, despite the fact that the impurity ion is different in our simulations. In regard to the energetic interactions, it appears that the results are not very sensitive to the details of the atomic interactions. In regard to the entropy, if the primary contribution of entropy in the binding process is configurational, the entropy of binding should be consistent across different cations in the same system, provided that the association mode is the same, e.g. substitutional or interstitial.

Taking the calculated entropy and volume of binding, we can obtain estimates of the binding energy between cation and vacancy along a mantle adiabat. We use a pressure profile generated with PREM (Dziewonski and Anderson), and assume a uniform adiabatic temperature gradient of 0.4 K/km, fixing the temperature at 400 km depth at

1830 K as in (Katsura et al.). The results are shown in Figure 2.5, for different values of the binding enthalpy at zero pressure. Increasing temperature and pressure have opposite effects on the binding energy, but along an adiabat, the effect of pressure is stronger.

Hence, the binding energy is predicted to increase significantly with increasing depth in the lower mantle. This implies that, for periclase of a given composition, fewer unbound cation vacancies will be available at greater depths, and that cation diffusion rates will be correspondingly slower.

Figure 2.5: Binding energy along a mantle adiabat, assuming a 0.4 K/km temperature gradient and PREM. Estimated binding energies obtained by a fit to the data in Table 2.1.

Shaded region denotes 95% confidence interval. Dotted lines are projected binding energies given different binding enthalpies, along the mantle adiabat.

2.5.2 Diffusion

The self-diffusion coefficient for magnesium in MgO as a function of free vacancy concentration is of importance in understanding the transport properties of pure crystals. For each simulation listed in Table 2.1, we obtained the mean squared displacement as a function of simulation time (see Equation 3.1 and surrounding discussion).

3 Structure Diffusivity Relations at Grain Boundaries in

Polycrystalline MgO

3.1 Chapter Summary

Molecular simulations are carried out to address the structure and atomic diffusion at grain boundaries. We use an inherent structure approach, which maps each configuration in a molecular dynamics trajectory to the potential energy minimum (“inherent structure”) it would reach by a steepest descent quench. Dynamics are then decomposed into a combination of displacements within an inherent structure and transitions between inherent structures. The inherent structure approach reveals a simple structural picture of the grain boundary that is normally obscured by the thermal motion. We apply our methodology to polycrystalline MgO. Grain boundary atoms are identified as atoms that are undercoordinated in the inherent structure, relative to those in the perfect crystal. Our method enables the calculation of grain boundary diffusion coefficients without arbitrary assumptions about which atoms or spatial regions belong to the grain boundary, and the results are shown to be consistent with estimates from experiments. The inherent structure approach also enables the elementary steps in the diffusion process to be elucidated. We show that the process in MgO grain boundaries primarily involves vacancy hops, but that there is also significant motion of other nearby atoms during such a hop.

3.2 Introduction

Grain boundaries are interfacial regions separating crystals of the same phase. They influence many properties of polycrystalline materials, including electrical conductivity

(Bean et al.), thermal conductivity (Liu et al.), and strain resistance (Sayle). Because grain boundaries are regions of strongly enhanced atomic mobility, they play a particularly important role in diffusion-dependent material properties, including creep, chemical transport, sintering and ionic conductivity (Sutton and Balluffi).

Although grain boundary diffusion has been studied for more than 60 years

(Fisher), the atomic level mechanisms that enhance transport within grain boundaries remain poorly known. Information on the atomic structure of grain boundaries can be obtained using high-resolution scanning TEM, but atomic-scale resolution is generally not obtainable in experimental studies of grain boundary diffusion. Instead, such studies typically rely on macroscopic evidence for enhanced transport near a grain boundary, and the data are analyzed in terms of simple models that treat the grain boundary as a homogeneous slab (of unknown width) (Inderjeet Kaur. et al.; Marquardt, Ramasse, et al.). Quantum-mechanical simulations have been used to predict possible grain boundary diffusion pathways and their energy barriers (Tschopp et al.; Verma and Karki; McKenna and Shluger), but do not address the dynamics of atomic hops. Molecular dynamics simulations with parameterized force fields have been used to model atomic motions in systems with grain boundaries, (Verma and Karki; McKenna and Shluger; Vincent-

Aublant et al.) including some simulations that investigate diffusion mechanisms

(Sørensen et al.; Landuzzi et al.; Karakasidis and Meyer). However, to determine a diffusion coefficient at grain boundaries it is necessary to specify which atoms or spatial

26 regions within the simulation belong to a grain boundary, which is not well defined.

Nonetheless, the identification of grain boundary atoms has been attempted by criteria based on the atom displacements (Mantisi et al.), the position in the initial structure(Karakasidis and Meyer), or the position in the instantaneous structure (e.g., within a slab of specified width centered on the grain boundary) (Landuzzi et al.). The first approach would tend to over-estimate diffusion at grain boundaries by ignoring immobile atoms at the grain boundary, and the second two approaches are based on assumptions about the grain boundary that are arbitrary.

Simple textbook pictures of grain boundaries show grains composed of perfect crystal lattices, separated by clearly identifiable grain boundaries (Figure 3.1a). However, such pictures are not realistic depictions of an instantaneous atomic structure, due to thermal motion of the atoms. Figure 3.1b shows the same polycrystalline structure as in

Figure 3.1a, but with random displacements of the atoms from their lattice sites to simulate the effects of thermal motion. In Figure 3.1b, the position and structure of the

27 grain boundary is not apparent, even though the deviation of each atom from its lattice position is relatively small.

In this paper, we show how the inherent structure (or “energy landscape”) method

(Stillinger and Weber) can be used to reveal the simple grain boundary picture that is ordinarily obscured by the thermal motion. The essence of the method is shown in Figure

3.1c – the goal is to find the idealized structure (open circles) that underlies an instantaneous configuration (closed circles). The potential energy of a system of N atoms is a 3N-dimensional function of the coordinates of the atoms. There will be a huge number of local minima of this potential energy function. The local energy minima correspond to atomic configurations with zero net force on each atom; each zero-net- force configuration is called an “inherent structure”. The deepest minima correspond to inherent structures that are perfectly crystalline (note there could be multiple perfectly crystalline structures, of different crystalline symmetry). Many other local minima come about when there is disorder in the system, and correspond to inherent structures with localized areas of disorder (polycrystalline systems) or with disorder throughout the system (liquids and glasses).

The inherent structure method is a rigorous method to describe the dynamics in thermal systems. Any instantaneous configuration in a molecular dynamics simulation

(which we will call an “instantaneous thermal structure”) can be unambiguously mapped to its corresponding inherent structure by a steepest descent energy minimization; this process is shown schematically in Figure 3.1c. Thus, the instantaneous thermal structure can be uniquely described in terms of (1) its corresponding inherent structure, and (2) the displacements of the atoms from this inherent structure. As a molecular dynamics

28 trajectory proceeds, the instantaneous thermal structure changes. For short time scales, the instantaneous thermal structure continues to map to the same inherent structure, while the displacements of the atoms from this inherent structure change; this motion is considered vibrational, in concurrence with the usual definition of vibrational motion as oscillatory motion about an energy minimum. For longer time scales, the instantaneous thermal structure maps to different inherent structures; this motion, corresponding to transitions between different inherent structures, is associated with atomic translations that lead to diffusion. Thus, without approximation, the dynamics of a system can be described in terms of vibrational motion within an inherent structure, and transitions between inherent structures. The inherent structure approach has been widely used to understand the properties of liquids and glasses, including the glass transition (Stillinger and Weber) and the mechanical response of glasses (Lacks and Osborne). The present work is, to our knowledge, the first application of the inherent structure method to study behavior at grain boundaries.

The present study focuses on grain boundary behavior in MgO. Metal oxides such as MgO have a wide range of technological applications, particularly in the form of thin nanocrystalline films (Bean et al.) within which grain boundaries may strongly affect the bulk properties. MgO is also a major constituent of Earth’s lower mantle, and its bulk and grain boundary diffusion properties are important for understanding mantle rheology

(James A Van Orman et al.), and the velocity and attenuation of seismic waves.(van Beest et al.)

3.3 Methods

We carry out molecular dynamics simulations of polycrystalline MgO. The molecular dynamics method determines the time evolution of a system of atoms that follow classical dynamics. From these atomic trajectories, properties such as local structure, dynamics and diffusivity can be evaluated.

We simulate a polycrystalline system with four grains. The grains are in the form of slabs that are infinitely wide in two dimensions, and approximately 40 Å thick in the third dimension. Periodic boundary conditions are used in all directions. We used four grains because simulations with fewer grains showed coupled behavior between the grain boundaries, due to the periodic boundary conditions.

The system was prepared by the following steps:

1. The initial simulation cell was created as follows. A cubic MgO lattice was constructed with Mg-O distance of 2.106 Å. This lattice was then rotated around the y axis. Periodic boundary conditions constrain the possible rotation angles; an angle of rotation of ~24° was chosen, which in combination with the simulation cell length in the x-direction, leads to a continuous structure with the periodic boundary condition. A second lattice was created by rotating the first lattice by 180° about the 푦 axis, and switching the magnesium and oxygen ion positions. These two lattices were then replicated and translated along the z axis. The resulting system of four tilt-axis grains has a total of 134,232 atoms, and is shown in Figure 3.2. There are 35 layers in the plane of the tilt axis (normal to Figure

2.2). The complete process used to create the system can be found in Section 6.1.2 in the appendix.

Figure 3.2: Initial setup of the simulated system. Four distinct grain boundaries exist.

The tilt axis is in the y direction, with a grain boundary angle of 42°. The z direction is normal to the grain boundary plane. Large spheres are oxygen ions, while small spheres are magnesium ions. Atoms are colored by their distance to their 6th nearest neighbor

(blue atoms have their sixth nearest neighbor at ~2.1 Å, red atoms have their sixth- nearest neighbor at ~3.5 Å.) The grain boundary angle between two adjacent grains, defined as the angle needed to rotate the lattice of one grain onto the other, is ~42°. The resulting grain boundary is a symmetric tilt boundary of 24° around [100].

2. The initial structure generated in step 1 has empty space between the grains. To remove this empty space, a molecular dynamics simulation at constant anisotropic stress was carried out, with stresses of 5 GPa normal to the grain boundary and 1 GPa in the other directions. The simulation was carried out at 300 K for 60 ps.

3. To bring the system to its desired temperature, a molecular dynamics simulation at constant anisotropic stress was carried out, with stresses of 2.5 GPa normal to the grain boundary and 0.5 GPa in the other directions. The simulation was carried out with the temperature ramped from 300 K to the desired temperature over 60 ps.

4. To bring the system to its zero-pressure equilibrium dimensions at the desired temperature, a molecular dynamics simulation at zero pressure and the desired temperature was carried out for 60 ps.

After the system was prepared according to this procedure, a long simulation was run at constant volume, with cell dimensions determined from step 4. During this simulation, we find the inherent structures that the system visits as a function of time.

Configurations from the molecular dynamics simulation are stored at intervals of 1 ns.

The inherent structure corresponding to a given configuration is found by carrying out a steepest descent energy minimization starting from this configuration, such that all atoms move “downhill” on the energy landscape until they reach a local minimum and all atoms are at zero-force positions. Note that these minimizations are carried out as a post- processing step, and do not affect the molecular dynamics trajectory.

We note that our steepest descent minimizations were carried out in single precision, and converged to minima corresponding to the inherent structures included in this paper. It is desirable to examine the normal mode eigenvalues to confirm that the inherent structures are in fact energy minima; all non-zero normal mode eigenvalues (i.e., all but the three eigenvalues that correspond to center-of-mass motion) would be positive at an energy minimum. Unfortunately, the calculation of normal mode eigenvalues is not possible for our very large system size (134,232 atoms). Therefore, we carried out a test of our procedure on a small system (3,168 atoms) with the same grain boundary structure, but only two grains. Within this small system, we confirmed that the inherent structures are in fact energy minima, with all relevant normal mode eigenvalues being positive. We note that when we carry out the steepest descent minimizations with lower tolerances and higher precision the minimization procedure runs into convergence problems, which we believe to be due to the very high dimensional space and the fact that the minimization algorithm proceeds by finite steps that can “inadvertently step over” a small barrier and end up in a different energy minimum.

The details of our simulation methodology are as follows. The atoms interact via interatomic forces described by a modified van Beest (van Beest et al.) potential adapted for MgO, as previously reported (Daniel J Lacks, Rear, et al.). Short range interactions are summed completely (note that the potential has a smooth cutoff at 5.5 Å), and

Coulombic interactions are calculated using the particle-mesh Ewald algorithm (Darden et al.). Temperature is controlled with a Nosé-Hoover thermostat (Nosé; Hoover) with a time constant of 2 ps. Constant-stress simulations are carried out with a Berendsen

33 barostat (H J C Berendsen et al.) with a compressibility factor 5.8x10-7 bar-1. The equations of motion are integrated with a time step of 1.5 fs.

The simulations are carried out with the GROMACS molecular dynamics software package (H.J.C. Berendsen et al.). Results are visualized with the VMD software package (Humphrey et al.), with rendering done via the Tachyon library backend. Python’s MDAnalysis library (Michaud-Agrawal et al.) is used in some of the analysis.

3.4 Results

Molecular dynamics simulations are carried out for the polycrystalline MgO systems at temperatures of 1300 ℃, 1500 ℃ and 1800 ℃. The simulations are carried out for durations on the order of 100 ns.

First, we consider the structure. For a perfect MgO crystal, each Mg atom has six nearest neighbor O atoms that vibrate around the equilibrium nearest-neighbor distance.

Our molecular dynamics result for the probability distribution of the distances to the six nearest O atoms from an Mg atom, at 1800 ℃, is shown in Figure 3.3a. The distribution has a peak at about 2.2 Å, which is close to the experimentally determined nearest- neighbor distance of 2.15 Å at 1738 ℃ (Hirata et al.). The probability distribution in

Figure 3.3a is very broad, due to the thermal motion of the atoms.

Figure 3.3:Probability distribution for the six nearest Mg-O distances around an Mg atom. (a) results for configurations during the molecular dynamics trajectory at 1800 °C.

(b) results for inherent structures obtained from the molecular dynamics trajectory at

1800 °C.

Importantly, Figure 3.3a shows only one peak. We know that there are two classes of Mg environments in our system – Mg atoms in the bulk, and Mg atoms at grain boundaries. These two classes are expected to have different probability distributions for the six nearest Mg-O distances. However, the thermal motions obscure this important

35 aspect of the structure, and thus the grain boundary structure remains hidden in these results.

We use the inherent structure approach to uncover the grain boundary structure that is hidden in Figure 3.3a. As described above, inherent structures are obtained by steepest-descent quench from instantaneous configurations during the molecular dynamics trajectory. Figure 3.3b shows the probability distribution of the distances to the six nearest O atoms from an Mg atom, obtained in this case from the inherent structures at 1800 ℃. The distribution has a narrow peak at 2.17 Å. However, there is also a second peak at 3.7 Å. The Mg atoms that are associated with this second peak are undercoordinated, in that they have fewer than six O neighbors at the nearest-neighbor distance. We use the position of the minimum in the probability distribution as the cut-off to define Mg atoms as fully coordinated or undercoordinated.

The undercoordinated atoms would be expected to be at the grain boundaries, but they could also be associated with vacancy defects within the crystal. To address this issue, we examine the spatial distribution of the undercoordinated atoms as a function of the z dimension. Results for this spatial distribution, for one inherent structure from the simulation at 1800 ℃, are shown in Figure 3.4. We see that all of the undercoordinated atoms are at the grain boundaries, with none occurring within the crystallites (for this particular configuration). Thus we can associate the undercoordinated atoms with the grain boundary. Figure 4 also shows that the grain boundary is approximately 10 Å wide.

The number of atoms identified as grain boundary atoms increases with increasing temperature, as shown in Figure 3.5. This result implies that the grain boundaries become wider with increasing temperature (even after correcting for thermal expansion). From Figure 3.5 we also see that the number of atoms in the grain boundary during each constant-temperature simulation remains approximately constant over time.

Figure 3.5: Fraction of the Mg atoms in the system that are undercoordinated. The curves are results for 1300 °C (blue, bottom), 1500 °C (orange, middle), 1800 °C (gray, top).

The nature of the dynamics in the grain boundaries depends on temperature.

Figure 3.6 shows the fraction of undercoordinated atoms at a given time that were also undercoordinated one nanosecond previously. At 1800 ℃, about half of the atoms in a grain boundary switch into or out of the grain boundary within one nanosecond (where the atoms in the grain boundary are taken as those that are undercoordinated). At

1500 ℃, about 20% of the atoms in a grain boundary change within a nanosecond. At

1300 ℃, the simulation did not become equilibrated during the preparatory simulations before the production run (as was the case for the higher temperature simulations), and thus the behavior changes with time as the system equilibrates and the grain boundaries become disordered.

Mean-squared displacements for Mg and O are determined as a function of time, and the results are shown in Figure 3.7. As expected, the mobility depends very strongly on temperature, and the mean-squared displacements are almost 1000 times greater at

1800 ℃ than at 1300 ℃. The mobility is significantly higher for Mg than for O, with

38 mean-squared displacements larger by a factor of about 5-10. The mobility is found to be essentially the same in the x and y directions, even though the structure of the grain boundary is anisotropic in these dimensions (the y direction, being the tilt axis, has open channels, while the x direction, being normal to the tilt axis, has a kite structure). The lower mobility in the z direction (normal to the grain boundary plane) is expected, as significant atomic motion in this direction would involve movement of atoms within crystallites.

Figure 3.7: One-dimensional mean-squared displacements as a function of time. Blue data is for the x-dimension (in the grain boundary plane perpendicular to the tilt axis), orange data is for the y-dimension (down the tilt axis), and gray data is for the z- dimension (perpendicular to the grain boundary plane). A dashed line with a slope of unity is given as a reference.

In the diffusive regime, the mean-squared displacement increases linearly with time, with the proportionality defining the one-dimensional diffusion coefficient.

2 ⟨(푥 − 푥0) ⟩ = 2퐷푥푡 (3.1)

Plotting the mean-squared displacement as a function of time on a log-log plot, as in

Figure 3.7, enables a test of whether the system is in the diffusive regime – the slope on the log-log plot is equal to one when the system is in the diffusive regime. Our results show that the mobility in the x and y dimensions appears to be in the diffusion regime for both Mg and O at 1800 ℃, and for Mg at 1500 ℃.

3.5 Discussion

The goal of our work is to improve the understanding diffusivity at grain boundaries. Our approach leads to: (a) unambiguous and well-defined estimates of diffusion coefficients at grain boundaries; (b) elucidation of elementary steps in the diffusion process as a vacancy-mediated process; and (c) prediction of a non-Arrhenius temperature dependence of the diffusion coefficients at grain boundaries.

3.5.1 Estimate of diffusion coefficients

Estimation of the diffusion coefficient at grain boundaries is complicated by the fact that atoms move in and out of grain boundaries, as described above. Thus, a method that tracks a fixed set of “grain-boundary atoms” has the flaw that the set of grain boundary atoms changes with time. Similarly, a method that tracks the motions of atoms within a certain volume, e.g. a slab of specified width centered on the grain boundary, would not in general capture all and only grain boundary atoms, nor would it account for

41 changes in the effective width of the grain boundary with temperature, pressure or composition.

Our inherent structure approach provides an unambiguous path to estimating diffusivity at grain boundaries. The mean-squared displacements shown above were obtained as the mean over all atoms in the system. These total mean-squared displacements are the weighted average of the mean-squared displacements of atoms in the bulk crystallites and the mean-squared displacements of atoms at the grain boundaries,

2 2 2 ⟨(푥 − 푥0) ⟩푎푙푙 = 푓푐푟푦푠푡⟨(푥 − 푥0) ⟩푐푟푦푠푡 + 푓푔푏⟨(푥 − 푥0) ⟩푔푏 (3.2)

where the weighting factors fcryst and fgb are the fraction of atoms in the crystallites and in grain boundaries, respectively. The mean-squared displacements for atoms in the bulk

2 crystallite, ⟨(푥 − 푥0) ⟩푐푟푦푠푡, are negligible, as demonstrated in Figure 3.8, which shows the mean-squared displacements for the subset of atoms that remain in the bulk crystallite for the entire simulation. Hence, to a good approximation, the mean-squared displacement of atoms in the grain boundary can be obtained as,

2 1 2 (3.3) ⟨(푥 − 푥0) ⟩푔푏 ≅ ⟨(푥 − 푥0) ⟩푎푙푙 푓푔푏

This approximation in going from (3.2) to (3.3) leads to a difference of less than 5% in the mean-squared displacement.

We can thus use (3.1) and (3.3) to obtain estimates of the diffusion coefficient for atoms at grain boundaries, without tracking a particular subset of atoms or accounting for the time this subset spends within or outside of a grain boundary. We can

42 straightforwardly use (3.3) because fgb has a well-defined constant value (see Figure 3.5) despite the fact that the

Figure 3.8: Mean squared displacements as a function of time, for Mg atoms at 1800 °C.

Results are shown for all Mg atoms, and for Mg atoms that spent the entire simulation only in the bulk crystal (i.e., not at a grain boundary).

particular atoms in the grain boundary change significantly with time (see Figure 3.6). In this way we obtain the estimates for the diffusion coefficients at grain boundaries at

-11 2 1800 ℃, in the directions parallel to the grain boundaries, to be DMg,gb=4 x 10 m /s and

-11 2 DO,gb=0.6 x 10 m /s, for Mg and O, respectively.

To validate our simulation results, we compare with available experimental data.

Diffusion at grain boundaries is difficult to address experimentally. However, for O, data exists for the product of the grain boundary width () and the diffusion coefficient at the

-21 3 grain boundary (DO,gb); for MgO at 1800 ℃, experiments find DO,gb = 2.2x10 m /s

(Frost and Ashby). Considering ~1 nm (Figure 3.4b), the experimental estimate for DO,gb is 0.2 x 10-11 m2/s, which is within a factor of 3 of the value determined from our simulations (0.6 x 10-11 m2/s).

To our knowledge, no experimental measurements of Mg grain boundary diffusion in MgO exist. However, we can attempt to validate our result two ways. First, data is available for diffusion of Ni2+ defects in MgO bicrystals (Wuensch and Vasilos,

“Grain-Boundary Diffusion in MgO”). When combined with diffusion data for single crystals (Wuensch and Vasilos, “Diffusion of Transition Metal Ions in Single-Crystal

MgO”), and considering as above that the grain boundary width is ~ 1 nm, an experimentally-derived estimate for the grain boundary diffusion coefficient DNi,gb is 7.4 x 10-10 m2/s. Second, we can compare to results for Mg diffusion in single crystals, extrapolated to high vacancy concentrations comparable to those within the grain boundaries. In our simulations, each Mg and O grain boundary atom is undercoordinated, such that the Mg vacancy concentration in the grain boundary is ~1/6. In an MgO crystal with vacancies, the Mg diffusion coefficient can be calculated as DMg = xvDvf, where xv is the vacancy concentration, Dv is the cation-vacancy diffusion coefficient, and f is a correlation coefficient with a value close to one. Using the experimental estimate Dv =1.1

-10 2 x 10 m /s for MgO at 1800 ℃ (Sempolinski and Kingery), and xv = 1/6, we obtain the

-11 2 -11 2 estimate DMg ~ 1.4 x 10 m /s. Our simulation result for DMg,gb (4 x 10 m /s) is

-11 2 -10 bracketed by these two experimental estimates for DMg,gb (1.4 x 10 m /s, 7.4 x 10 m2/s).

3.5.2 Elucidation of elementary steps in the diffusion process at grain boundaries

Our approach enables us to elucidate the mechanism for diffusion at grain boundaries. As described above, without approximation, the dynamics of a system can be described in terms of vibrational motion within an inherent structure, and transitions

44 between inherent structures. The transitions between inherent structures are what lead to diffusion dynamics. We can thus examine the structural rearrangements associated with these transitions to determine the elementary steps in the diffusion process.

Figure 3.9: Example of a transition between two inherent structures during a simulation

We discuss here one particular example. Figure 9 shows relevant atomic positions before and after a transition between inherent structures, at 1300 ℃. In this case, the structural change corresponds mainly to an oxygen atom making a jump across the grain boundary, from a stable configuration on the upper grain to a stable configuration on the

45 lower grain. Before the jump, the oxygen atom is coordinated to 3 Mg atoms in the plane shown (the oxygen is also coordinated to other Mg atoms in different planes); these Mg atoms are denoted Mg1, Mg2, and Mg3. After the jump the oxygen atom is still coordinated to Mg3, but now it is coordinated to Mg4 and Mg5 rather than Mg1 and

Mg2.

The effect of this elementary diffusive step is to move the local position of the grain boundary by one layer. The oxygen atom labeled A had been fully coordinated within the crystallite before this transition, but now has become an undercoordinated atom that is part of the grain boundary. On the other side, the oxygen atom labeled B changes from being undercoordinated and part of the grain boundary to becoming fully coordinated and part of the crystallite.

We see that the transition can be described as a vacancy jump. However, the transition is not simply a jump of one atom from one lattice site to another lattice site.

Rather, there is also significant motion of other atoms during the transition, as shown in

Figure 3.9c.

The grain boundary can be thought of as a narrow region with very high vacancy concentrations. As shown above, the Mg diffusion coefficient is similar to the experimental value for diffusion in a crystal, extrapolated to a similar vacancy concentration (xv = 1/6). This result suggests that the energy barrier to a vacancy jump is similar in the grain boundary and within the crystal.

3.5.3 Possible non-Arrhenius diffusive behavior at grain boundaries

The inherent structure approach has been used to elucidate the rapid non-

Arrhenius change in properties (e.g., diffusivity, viscosity) in glass-forming liquids as the glass transition is approached (Sastry, Debenedetti, and Stillinger; Debenedetti and

Stillinger). If temperature decreases and barrier heights remain the same, dynamics will slow down with an Arrhenius (exponential) temperature dependence, due to the decreased rate by which the system can surmount energy barriers separating inherent structures.

However, decreases in temperature move the system to regions of the energy landscape characterized by deeper inherent structures; these deeper inherent structures are separated by higher barriers. Thus, as the temperature decreases, the dynamical properties change more rapidly than predicted by an Arrhenius dependence, because there is not only less energy to enable the system to surmount energy barriers, but also the energy barriers become larger as well.

We suggest that an analogous non-Arrhenius dependence may occur for diffusivity at grain boundaries. Figure 3.10 shows that as the temperature decreases, the grain-boundary systems move to regions of the energy landscape with deeper energy minima. These results are analogous to those found previously in glass-forming liquids

(Sastry, Debenedetti, Stillinger, et al.). We expect that the deeper energy minima will have larger barriers separating them. Thus we predict, in analogy to the glass-forming liquid results (Sastry), that there would be a non-Arrhenius dependence of the diffusion coefficient. We note that the present simulations do support this prediction, in that the drop-off in mean-squared displacement from 1500 ℃ to 1300 ℃ is much greater than the drop-off from 1800 ℃ to 1500 ℃ (i.e., the difference is much greater than expected from

47 an Arrhenius dependence). Our future work will include more simulations in order to address this prediction quantitatively.

4 A Molecular Dynamics Study of Grain Boundary Diffusion in

MgO

4.1 Chapter Summary

Molecular dynamics simulations are carried out on polycystalline periclase

(MgO) to determine the structure and diffusivity at grain boundaries for pressures and temperatures relevant to Earth’s mantle. Diffusion coefficients for Mg and O As temperature increases, the grain boundary structure becomes more disordered, with more ions having incomplete coordination and the system occupying regions of the energy landscape with shallower energy mimima. As pressure increases, in contrast, the grain boundary structure becomes more ordered. The grain boundary diffusivity as a function of temperature and pressure can be understood in terms of these structural changes. At atmospheric pressure, the grain boundary diffusion coefficients for Mg and O extrapolate with increasing temperature to the values for the melt, indicating that the dynamics in the grain boundary are similar to those of a supercooled liquid. Just as in a supercooled liquid, diffusion in the grain boundary slows down with decreasing temperature for two reasons: there is less energy to surmount energy barriers, and the barriers are larger due to the more ordered structure. As pressure increases from zero pressure, the diffusivities first decrease sharply, due to the increase in energy barriers associated with the more ordered system, and then more gradually as pressure increases beyond ~4 GPa. At conditions relevant to Earth’s core-mantle boundary region (135 GPa, 3500 oC), no diffusion is observed after ~100 ns, and the diffusion coefficients are constrained to be <10-13 m2/s.

This upper limit is considerably smaller than values obtained from experimental

49 measurements at lower pressures and temperatures, suggesting that grain boundary transport at the core-mantle boundary is considerably less efficient than has previously been inferred.

4.2 Introduction

Grain boundaries are interfacial regions connecting a network of regular crystal lattices, where the crystal lattices are of the same phase (Lejček). The grain boundary regions generally have chemical and transport properties that are very different from the interior of the crystals. For this reason, grain boundaries can significantly alter the properties of polycrystalline materials, including their bulk chemical diffusivity (Dohmen and Milke) and their creep viscosity (Frost and Ashby).

Diffusion of atoms along grain boundaries is in general several orders of magnitude faster than diffusion within crystal lattices (Joesten). This rapid transport along grain boundaries can enable chemical transfer between physically separated mineral pairs (Dohmen and Milke; Eiler et al.), and potentially enable large-scale transport between major geochemical reservoirs such as the core and mantle (James A

Van Orman et al.; Hayden and Watson, “A Diffusion Mechanism for Core–Mantle

Interaction”). Grain boundary diffusion also plays a key role in many geophysical processes. It can control mantle viscosity under some conditions (Hirth and Kohlstedt), is an important mechanism in the attenuation of seismic waves (Karato and Spetzler), and contributes to the temperature dependence of seismic velocities (Karato and Wu).

Periclase (MgO, with FeO in solid solution) is thought to be the second most abundant phase in the lower mantle, and may be the most abundant phase in some regions, including in the thin, ultra-low velocity zones just above the core (Wicks et al.).

Periclase has lower creep strength than the other major minerals of the lower mantle (Lin et al.; Reali et al.), and hence may have an important influence on mantle viscosity.

Studies of periclase suggest that grain boundary diffusion may govern viscous creep

51 under some mantle conditions (James A Van Orman et al.; Frost and Ashby), as well as viscoelastic and anaelastic deformation (Webb and Jackson; Barnhoorn et al.). Because of its simple crystal structure and lack of phase transitions below 450 GPa (Karki et al.), periclase is also a useful model material for studying diffusion over a wide range of conditions.

Grain boundary diffusion in periclase has been studied experimentally (McKenzie et al.;

Osenbach and Stubican; James A Van Orman et al.; Hayden and Watson, “A Diffusion

Mechanism for Core–Mantle Interaction”; Wuensch and Vasilos, “Grain-Boundary

Diffusion in MgO”) but many uncertainties remain on the fundamental processes that are involved in grain boundary transport, and how the rates vary with temperature and pressure. Grain boundary diffusion is typically modelled as a thin, highly diffusing region of width 훿, with diffusion coefficient 퐷퐺퐵 and, for impurity atoms, a segregation coefficient, s, which represents the equilibrium fraction of impurity in the grain boundary.

In systems where volume and grain boundary diffusion both occur, experiments typically cannot disentangle the triple product 푠훿퐷퐺퐵 (Joesten).

Further insight on grain boundary diffusion can be obtained from molecular simulations. Ab initio simulations based on density functional theory (DFT) have been performed on MgO systems with grain boundaries on (Verma and Karki). These calculations provide important insight on grain boundary structure, and on the migration enthalpies for specific atomic transitions within the grain boundary, but cannot at present fully account for the entropic effects involved in grain boundary diffusion, as DFT calculation of all of the thermally available grain boundary conformations is computationally expensive. Kinetic monte carlo (KMC) simulations have also been used

52 to study grain boundary diffusion (Harding and Harris; Ma et al.), but these simulations require a specific grain structure to be assumed, and also require knowledge a priori of the attempt frequencies for each mode of migration.

In this paper, we use classical molecular dynamics, which enables large scale simulations for long simulation times, without supposing a predetermined grain boundary lattice from which atoms transition. Molecular dynamics simulations of grain boundary diffusion are computationally intensive, even using simple pairwise atomic interactions.

To simulate a polycrystalline material requires large systems containing many atoms, and long run times are necessary to study the migration of atoms in the diffusive regime.

Simulations are performed on a polycrystalline MgO system to determine Mg and O grain boundary diffusion coefficients over a range of pressures and temperatures corresponding to conditions within Earth’s mantle. We show that the dynamics of grain boundary diffusion are related to grain boundary structure, and are analogous to the behavior of liquids near the glass transition temperature.

4.3 Methods

In the molecular dynamics simulations of polycrystalline MgO, the key considerations are how to: (a) define and construct the polycrystalline system; (b) model the atomic interactions; (c) simulate the system dynamics and equilibration; and (d) analyze the results to obtain meaningful physical insight. We describe below our approaches in these regards.

4.3.1 The polycrystalline system

The systems are modeled with periodic boundary conditions in all directions, in order to remove surface effects. However, constructing polycrystalline systems that

53 satisfy periodic boundary conditions is challenging. Systems with more than two grains are needed, because the periodic boundary conditions cause coupled grain boundary motion (a nonphysical result) if only two grains are used. For this reason, we simulate a system with four grains, where each grain is an infinite slab oriented with the grain boundary perpendicular to the z axis (see Figure 4.1).

Figure 4.1: Representative equilibrated structure. View is normal to the angle of rotation of the grain boundaries. Red spheres are oxygen ions, and white spheres are magnesium ions.

In general five parameters are required to describe the relative positions and orientations of the crystallites at the grain boundary (Saylor et al.). Here, we address a symmetric tilt boundary system with the axis of rotation in the [1 0 0] plane of both neighboring crystallites, such that the grain boundary is described by only one angle, 휃퐺퐵, without translational offset between the lattices. We choose to address high-angle grain boundary structures (tilt angle between the grains >15o) because they predominate in

54 equilibrated polycrystalline aggregates, including periclase (Saylor et al.) and olivine

(Marquardt, Rohrer, et al.), and are therefore expected to be more relevant to grain boundary diffusion in Earth’s interior. High-angle grain boundaries are also convenient to study because diffusion is (1) faster (Sutton and Balluffi; Mohammadzadeh and

Mohammadzadeh); (2) less dependent on the particular value of the angle (Stubican and

Osenbach; Le Claire), and (3) less anisotropic compared to low-angle grain boundaries.

Preserving the lattice structure across periodic boundaries requires certain restrictions on the cell structure. We generate simulation cells with 푛푠푖푑푒 atoms on each

푛푗 side. Generation of a grain boundary with angle 휃퐺퐵 requires that tan 휃퐺퐵 = , where 푛푗 푛푘 and 푛푘 are integers such that 푛푗 ≤ 푛푘 ≤ 푛푠푖푑푒 and the sum of 푛푗 and 푛푘 is even. The simulation cell length in the direction normal to the axis of rotation and within the grain

2 2 boundary plane is restricted to a multiple of 푎√푛푗 + 푛푘, where 푎 is the lattice constant.

The simulation cell length in the direction normal to the grain boundary plane, 푙푧, is

푛 휃 푙 = 4 (2 ∙ 퐼푁푇 ( 푠푖푑푒) + 1) 푎 cos 퐺퐵 (4.1) 푧 4 2

where the INT function truncates the argument to the nearest integer value. This procedure ensures that each grain has an odd number of ions in this direction, and thus has ions of opposite charge as nearest neighbors across the grain boundary. The box length in the direction parallel to the rotation axis of the grain boundary is constrained to

푛 be (2 ∙ 퐼푁푇 ( 푠푖푑푒) + 1) 푎, which gives an initial structure with four identical grains with 4 parallel grain boundaries, as shown in Figure 4.1.

The present simulations use nside=35, nj=21, and nk=47, which gives 휃퐺퐵≈41.85°. Since grain boundary diffusion tends to become faster as the tilt angle increases, and 휃퐺퐵 ≤ 45° due to symmetry, our simulations are likely to be similar to those of the angle for which maximum grain boundary transport occurs (Osenbach and Stubican; Stubican and

Osenbach; Upmanyu et al.). Although the grain boundary tilt angle is set only as an initial condition, it remains consistent throughout the simulation. Because the simulations have an inherent periodicity, they are coincident site lattices (CSL), grain boundary systems where lattice points for the opposing grains coincide, which gives them unique properties.

It is known that CSL lattices with low values of Σ (e.g. 3,5,7) have special properties including a low interfacial energy. Our grain boundary interfaces, which were able to rearrange as necessary, are within 5° of a Σ5(310)[001] boundary, but do not rearrange to form this boundary (the boundaries are Σ1325(13 34 0)[001]). Our results, therefore, should be representative of a general grain boundary, rather than a special configuration that may have distinct (slower) diffusional transport properties

(Shvindlerman and Straumal; Aleshin et al.).

For accurate representation of a real material, each grain must be large enough so that its interior behaves like a bulk crystal. This condition requires that a large simulation cell be used. The simulation cell we use includes 134,232 atoms, which leads to a center- to-center distance between the grain boundaries of approximately 4 nm; this length can be interpreted as a grain size. For the conditions we report, each grain boundary is independent of the other grain boundaries, and atoms that participate in one grain boundary are unable to participate in any other over the course of our simulation.

4.3.2 Interatomic forces

There are two classes of approaches to modeling interatomic forces. First principles (ab initio) approaches solve (with approximations) the Schrödinger equation for the electrons to determine the forces on the nuclei, while force field approaches parameterize the forces to simple functional forms. The first principles approach has the advantage that it can account for local atomic environments that differ from those used in the force field parameterization process; these local environment effects could be significant at grain boundaries. However, the first principles approach has the disadvantage of very high computational intensity, which currently limits these simulations to system sizes of hundreds of atoms and durations of tens of picoseconds.

The force field approach has the advantage of being much less computationally intensive, such that much larger systems can be addressed for much longer times. As described above, very large systems (~105 atoms) are needed to accurately model the effects of grain boundaries, and long simulation times (~100 ns) are needed to address diffusion in solid systems. For these reasons, we choose to use the force field approach in this work.

Both ab initio (Karki et al.) and experimental methods (Zha et al.) have demonstrated that the stress-strain relationship for MgO shows a clear Cauchy violation, and cannot, in general, be described with pairwise interactions that depend only on distance (“central forces”). This led to studies of bulk properties by the variational induced breathing method, which is a type of shell model that is more computationally expensive than a central-forces model (Inbar and Cohen; Ita and Cohen). However, when compared to experiments, a shell model (which can account for the Cauchy violation) did not improve the accuracy of material property calculations over a simpler central-forces

57 model (Landuzzi et al.). For this reason, and the reduced computational intensity, we use a central forces model here.

We model interatomic forces with a modified Van Beest potential (van Beest et al.), which is based on pairwise interactions parameterized by a simple functional form.

Parameters for the potential were derived by fitting to a mix of ab initio and experimental results (Daniel J Lacks, Rear, et al.; van Beest et al.). The potential has previously been used to model silicate and oxide liquids (Daniel J. Lacks et al.; Daniel J Lacks, Rear, et al.; Zhang et al.), and a preliminary study of MgO grain boundaries (Riet et al.)..

4.3.3 Simulation methodology

A series of molecular dynamics simulations is used to first equilibrate the system and then to determine system properties. All simulations are run at constant temperature, with the temperature controlled by a Nosé-Hoover thermostat (Nosé; Hoover). Some simulations used for system equilibration are also carried out at constant stress, with the stress regulated through a Berendsen barostat (H J C Berendsen et al.). The simulations used to determine system properties are carried out at constant strain rather than constant stress. The equations of motion are integrated using the leap-frog method with a time step of 1.5 fs. All simulations are carried out with the Gromacs 5.1 software package (H.J.C.

Berendsen et al.).

The initial structure we generate using the procedure described in Section 2.1 is not relaxed. To allow the system to relax without destroying the grain boundary structure, we perform a sequence of short molecular dynamics equilibration runs, prior to a long simulation that is used to determine the structural and transport properties of the grain boundaries. First, we carry out a simulation at T= 300 K for 60 ps, with an applied stress

58 of 5 GPa normal to the grain boundary and applied stresses of 1 GPa in the other directions to maintain the structure. The system temperature is then stepped to the desired temperature over the course of another 60 ps simulation, with the same applied stresses.

The final preparatory simulation is run at the desired pressure (rather than the applied stresses) and the desired temperature for 60 ps. After this preparation, the system is run under constant volume (NVT) conditions for approximately 100 ns. We use this final

NVT simulation to determine structural and transport properties.

4.3.4 Property analysis

A challenge with molecular dynamics simulations is how to obtain physical insight from the trajectory results. This issue is particularly relevant in the present study, because the greatly enhanced transport near the grain boundary creates spatially heterogeneous diffusional properties within the polycrystalline system. The most straightforward analysis methods would give overall properties of the system, rather than separate properties within the crystallites and in the grain boundaries. More advanced analysis methods are needed in order to estimate properties in the different regions.

As we showed previously (Riet et al.), the properties of polycrystalline systems can be understood using an energy landscape analysis, which strips away thermal vibrational displacements to give a clearer rendering of the positions of grain boundaries.

This analysis maps instantaneous configurations from the molecular dynamics trajectory to the nearest local energy minimum. The mapping is a post-processing step (i.e., it does not affect the molecular dynamics trajectory), and is carried out by a steepest descent energy minimization starting from configurations saved during the molecular dynamics trajectory. A sampling of energy minima is obtained at 1 nm intervals in the molecular

59 dynamics trajectory, and we characterize their properties. We analyze the nearest- neighbor distributions at the energy minima to find which atoms are fully coordinated with 6 nearest neighbors, and which atoms are undercoordinated, using a nearest neighbor search with the MDAnalysis python package (Michaud-Agrawal et al.). We have previously shown that the undercoordinated atoms are spatially associated with the grain boundaries, and that undercoordinated atoms within the interior of the crystals, due to the presence of crystalline point defects, are negligible (Riet et al.). We also characterize the atomic packing structure in the grain boundaries by the potential energy at the energy minima – lower energies imply a more ordered packing at the grain boundary that facilitates favorable atomic interactions.

The central objective of our simulations is to determine the diffusion coefficients of atoms at the grain boundaries. Because grain boundaries move with time, and atoms move in and out of grain boundaries, we cannot simply define particular “grain boundary” regions of the simulation box, or a distinct set of grain boundary atoms, and track the diffusion behavior of these regions or these atoms. Our approach to overcoming this issue is based on our finding (Riet et al.) that although the particular atoms at the grain boundaries change with time, the overall fraction of undercoordinated atoms in the grain boundaries, g, remains constant (to within limited fluctuations). In this way, grain boundary diffusion coefficients, 퐷퐺퐵, can be determined from the overall and volume diffusion coefficients, 퐷푡표푡푎푙 and 퐷푣표푙, with the modified Hart equation (Hart; Dohmen and Milke):

퐷푡표푡푎푙 = 푔 ⋅ 퐷퐺퐵 + (1 − 푔) ⋅ 퐷푣표푙 (4.2)

The total diffusion coefficient can be obtained from the mean squared

2 displacement (MSD), 〈푥 − 푥0〉 , over a duration 푡 by the Einstein diffusion relation

(Einstein),

〈푥−푥 〉2 퐷 = 0 (4.3) 푡표푡푎푙 6푡

We also analyzed trajectories of atoms that remain within the interior of crystals

(away from the grain boundaries) for the entire simulation, and found that these atoms undergo negligible diffusion in the timescales of our simulations. Thus, 퐷푣표푙 ≈ 0, and the following approximation for the grain boundary diffusion coefficient is accurate:

퐷 1 〈푥−푥 〉2 퐷 = 푡표푡푎푙 = 0 (4.4) 퐺퐵 푔 푔 6Δ푡

The Einstein diffusion relation (Equation 2.3) is only relevant for timescales where there is significant diffusion in the system (the diffusive regime); at shorter timescales the dynamics are dominated by atoms rattling in cages created by neighboring atoms, and the Einstein relation does not hold. We confined our analysis to conditions where the dynamics were sufficiently fast that the diffusive regime was reached on the timescale of the simulations; we determined whether the system is in the diffusive regime by examining whether the MSD followed power law behavior with respect to time, with a power law exponent of 1.

It is well known that grain boundaries can migrate with time (Rupert et al.;

Janssens et al.). Grain boundary migration occurs in some of our simulations, and in some cases two grain boundaries annhilate each other, leading to a two-grain (rather than four-grain) system. For determining diffusion coefficients, we only use data from the times where four grains exist in the system, and where grain boundary movement is small.

We calculate the overall diffusion coefficient from least squares estimation of the logarithm of the mean squared displacement as obtained by gromacs’ MSD utility. This recasts the slope-only linear regression into an intercept-only model, and allows estimation of the diffusion coefficient for small diffusivities without allowing negative diffusivities, which would be nonphysical.

2 ln(〈푥 − 푥0〉 ) − ln(Δ푡) = ln (6퐷) (4.5)

4.4 Results

Molecular dynamics simulations were carried out for polycrystalline MgO system at pressures between zero and 135 GPa, and temperatures between 1200 and 3500 °C. We identify relationships between grain boundary structure and diffusion, and how these properties change with temperature and pressure.

4.4.1 Structure

We characterize the structure in terms of the distribution of local ionic environments within our MgO system. Ions within the grain have six nearest-neighbor ions at identical average distances; the nearest-neighbor distance in the grain increases slightly with increasing temperature (due to thermal expansion), but decreases more strongly with increasing pressure over the ranges of interest. At the grain boundary, the ions do not have this crystalline environment. The ions that surround a central ion are at a distribution of distances, and are likely to have fewer than six nearest-neighbor ions (as defined by an appropriately chosen cutoff distance for “nearest-neighbor”).

The structural picture described above applies to the time-averaged positions of the ions in the material. In contrast, the instantaneous structure, based on atom positions at any point in time, differs significantly from this picture due to thermal motion. We

62 therefore analyze the structures at the energy minima (as described in section 4.3.4), which effectively strips away the thermal motion to reveal a physically-relevant picture of the structure.

Results for the distribution of distances from a central ion to its sixth-closest ion

(at the structures corresponding to the energy mimima) are shown inFigure 4.2. The first peak is the case where the sixth-closest ion is at the same distance as the five nearest ions; here, the central ion has its full complement of six neighbors. The second peak is the case where the sixth-closest ion is significantly further away than the nearest ions; here, the central ion has only five or fewer neighbors, and we refer to it as undercoordinated. In our simulations, the ions within the interior of the crystallites always have the full complement of six neighbors, and the undercoordinated ions are only found in the grain boundary region.

Significant changes in the local environment at the grain boundary occur with changes in temperature and pressure. First, the area under the peak for the undercoordinated ions decreases with decreasing temperature and increasing pressure, indicating that the grain boundary region becomes more ordered at decreasing temperature and at increasing pressure. Second, the sixth-closest ion distance decreases more strongly with increasing pressure for the undercoordinated ions than for the fully coordinated ions, indicating that the grain boundary is more compressible than the crystallites; for example, when

64 compressed from 0 GPa to 135 GPa, the sixth-closest ion distance decreases by 15% for the fully coordinated ions but by 25% for the undercoordinated ions.

The fraction of undercoordinated ions, as a function of temperature and pressure, is shown in Figure 4.3. As noted above, the fraction of undercoordinated ions increases with decreasing temperature and with increasing pressure. At low pressures, a temperature decrease of a few hundred degrees has a similar effect as a pressure increase of a few gigapascals.

Figure 4.3: Fraction of undercoordinated atoms as a function of (a) temperature, (b) pressure.

A lower fraction of undercoordinated ions implies more order in the region near the grain boundary, with more ions having the full complement of nearest-neighbor bonds. This effect is evident in the results for the average potential energy at the energy minima visited by the system. As shown in Figure 4.4, as temperature decreases the system visits deeper energy minima, corresponding to more ordered structures in which more ions have the full complement of nearest-neighbor bonds.

Figure 4.4: Potential energy of the energy minima visited by the polycrystalline system, at

P = 0 GPa. In comparison, the energy of the single crystal is shown (which increases with temperature due to thermal expansion).

The differentiation between bulk and grain boundary structure decrease as the pressure increases, as shown in Figure 4.5 for one particular grain boundary. At P=0 GPa, the density in the grain boundary is about 27% smaller than that within the crystallite; at

P=10 GPa it is about 14% smaller; and at P=50 GPa it is only about 7% smaller. This

66 change is due to the higher compressibility of the grain boundary in comparison to the crystallite. We also note that the width of the grain boundary decreases with increasing pressure from approximately 1.5 nm at 0 GPa to 1.0 nm at 10 GPa, to 0.75 nm at 50 GPa; this change is due to the smaller number of undercoordinated ions at increasing pressures, as shown in Figure 4.3.

Figure 4.5: Density near the grain boundary at three different pressures. The results at

P=0, 10 and 50 GPa are at temperatures of 1800, 2300 and 2000 °C respectively.

4.4.2 Diffusion

The grain boundary diffusion coefficients were determined using Equation 4.3, and results are presented in Table 1. At pressures higher than ≈10 GPa, the dynamics become so slow that the system does not reach the diffusive regime in the timescale of the simulations, and we are unable to determine diffusion coefficients.

Table 4.1: Overall diffusion coefficients, grain boundary widths, and grain boundary fractions from simulations. Diffusivity is overall diffusivity as obtained by a fit to the

Einstein diffusion equation. The values in parenthesis are 1σ standard errors in the last significant figure.

log10 Mg log10 Oxygen Time GB Pressure Diffusion Diffusion GB mol Temp [°C] Elapsed Width [GPa] Coefficient Coefficient Fraction [ns] [Å] [m²/s] [m²/s]

1300 0 197.1 <-13.3 <-13.3 0.11 6.1

1500 0 73.5 -11.8(3) -12.8(5) 0.13 8.1

1500 25 159 <-13.2 <-13.2 0.12 5.03

1600 0 99.9 -11.4(4) -12.3(9) 0.13 8.08

1700 0 99.9 -10.9(2) -11.9(3) 0.13 8.4

1800 0 87 -10.50(5) -11.3(3) 0.14 8.88

1800 0.5 77.2 -10.7(1) -11.5(5) 0.13 8.48

1800 1 49.8 -10.80(6) -11.7(2) 0.13 8.19

1800 2 24.6 -11.0(1) -11.9(3) 0.12 7.58

1800 3 89.2 -11.2(2) -12.2(10) 0.12 7.38

1800 25 116.7 <-13.1 <-13.1 0.12 4.58

1800 50 143.7 <-13.2 <-13.2 0.08 4.98

1900 0 44 -10.10(7) -10.8(2) 0.12 9.61

2000 0 21.6 -9.80(3) -10.40(8) 0.15 10.28

2000 1 31.5 -10.10(4) -10.8(1) 0.14 9.32

2000 2 39.3 -10.30(3) -11.1(2) 0.13 8.41

2000 4 57 -10.70(6) -11.6(2) 0.12 7.54

2000 8 103.2 -11.2(2) -12.5(6) 0.11 6.51

2000 25 53.4 <-12.7 <-12.7 0.12 4.46

2000 50 130.2 <-13.1 <-13.1 0.08 4.84

2000 75 147.6 <-13.2 <-13.2 0.06 3.66

2200 25 53.1 <-12.7 <-12.7 0.12 4.63

2200 50 78.3 <-12.9 <-12.9 0.09 4.35

2200 75 90 <-13.0 <-13.0 0.09 3.72

2200 105 108 <-13.1 <-13.1 0.05 3.41

2300 2 117 <-13.1 <-13.1 0.15 10.58

2300 4 100 <-13.0 <-13.0 0.13 8.71

2300 6 48.3 -10.10(3) -11.00(7) 0.12 7.86

2300 8 87.6 <-13.0 <-13.0 0.11 7.02

2300 10 99.9 -10.6(1) -11.7(5) 0.11 6.75

2300 12 69.8 -10.8(1) -12.0(5) 0.1 6.27

2400 75 100 <-13.0 <-13.0 0.07 3.68

2400 105 84 <-12.9 <-12.9 0.05 3.86

2400 135 37.8 <-12.6 <-12.6 0.05 3.8

2600 8 1.2 -9.5(5) -10.20(8) 0.12 8.59

2600 12 13.7 -10.0(1) -10.9(2) 0.1 7

3000 105 98.8 <-13.0 <-13.0 0.05 2.53

3000 135 52.2 <-12.7 <-12.7 0.05 2.94

3500 135 49.2 <-12.7 <-12.7 0.05 2.78

The diffusion coefficient is found to be larger for magnesium than for oxygen.

This result is understandable in that the magnesium ion is smaller than the oxygen ion – the ionic radii of the magnesium and oxygen ions is estimated as 0.86 Å and 1.26 Å, respectively (Shannon) – and thus it can more easily pass through free volume to enable diffusion. The ratio of the magnesium and oxygen diffusion coefficients changes with temperature and pressure – the ratio decreases with increasing temperature, but increases with increasing pressure. These dependences can be understood in terms of the relative

69 sizes of the ions. In regard to temperature, since the oxygen ion is larger, it has a higher activation energy to move through free volume, and thus its diffusion coefficient will increase more strongly with increasing temperature. In regard to pressure, since the oxygen ion is larger, its motion will be restricted more strongly as free volume decreases, and thus its diffusion coefficient will decrease more strongly with increasing pressure.

Similar behavior has been observed in simulations of liquid systems (Daniel J Lacks,

Rear, et al.) and in experiments on single crystal MgO (Van Orman and Crispin).

The temperature dependence of the diffusion coefficients at zero pressure (essentially atmospheric pressure) is shown in Figure 4.6. As expected, the diffusion coefficients increase with increasing temperature. The results appear to follow an Arrhenius relation; but as we discuss in Section 4.2, we believe the temperature dependence of the diffusion coefficients is more complex than it may first appear. As temperature increases, the diffusion coefficients approach the values for MgO melts in simulations with the same force field (Daniel J Lacks, Rear, et al.).

The diffusion coefficients are shown as functions of pressure in Figure 4.7. The diffusion coefficients decrease strongly with increasing pressure. We ran simulations at

25 GPa, 50 GPa, 75 GPa, 105 GPa, and 135 GPa at temperatures of 2200 °C and higher

(up to 3500 °C at 135 GPa) that did not reach the diffusional regime after 100 ns. These simulations place an upper limit of 10-13 m2/s for the Mg and O diffusion coefficients at these conditions.

4.5 Discussion

4.5.1 Comparison with experimental data

It is difficult to estimate grain boundary diffusion coefficients from experiments, and only a few studies have investigated grain boundary diffusion in MgO. Each of these studies has involved a polycrystalline MgO sample coupled to an environment enriched in isotopic magnesium and/or oxygen, or another cation such as Ni or Cr. After heating to a high temperature for a period of time, the distribution isotopes or cation dopant in the polycrystalline MgO sample is determined, and fit to a model to obtain information about the grain boundary diffusivity. The grain boundary diffusivity itself usually cannot be directly inferred using these methods. Most experiments have been performed in a kinetic regime, often denoted “type B”, e.g. (Dohmen and Milke), in which both grain boundary

푠훿퐷퐺퐵 and volume diffusion occur simultaneously. In these cases it is the product 0.5, where (퐷푣표푙) s is the segregation coefficient that describes the distribution of the tracer species between the grain boundary and bulk crystal (with 푠 ≅ 1 for Mg or O isotopes), that is obtained from the fit. Then, experimental values for 퐷푣표푙 can be used to back out the product

푠훿퐷퐺퐵, which cannot be separated further. Several experimental investigations have addressed grain boundary diffusion in MgO this way (Wuensch and Vasilos, “Grain-

Boundary Diffusion in MgO”; McKenzie et al.; Osenbach and Stubican). One study, by

Hayden and Watson (Hayden and Watson, “A Diffusion Mechanism for Core–Mantle

Interaction”), studied diffusion of several siderophile elements through polycrystalline

MgO using metal sinks placed within and/or at the edges of the samples. Because of the low solubility of the siderophile elements in MgO, diffusional transport in these

72 experiments was inferred to have occurred only along grain boundaries, and estimates of the grain boundary diffusion coefficients were obtained from concentration profiles determined from metal sink particles distributed through sample.

The most relevant experiments to compare to our results, in regard to the quantitative estimation of the magnitude of the diffusion coefficient, are those of

McKenzie et al. (1971). The experiments of Wuensch and Vasilos (1964), Osenbach and

Stubican (1983), and Hayden and Watson (2007) address the diffusion of other elements, which cannot be quantitatively compared to diffusion of Mg or O. The experiments of

Van Orman et al. (2003) examined Mg and O grain boundary diffusion, but were performed at high pressures (15-25 GPa), where water adsorbed to the samples from air during loading likely led to hydrogen segregation at the grain boundaries. The presence of hydrogen in these experiments is likely to have affected the grain boundary structure and thus the quantitative values of the grain boundary diffusion coefficients. We note, however, that these other experiments likely give useful insight into the temperature and pressure dependences of grain boundary diffusion, and will be discussed in that context.

훿퐷퐺퐵 o McKenzie et al. (1971) report the product 0.5 for oxygen diffusion at 1700 C. There (퐷푣표푙) is wide variation in literature values of 퐷푣표푙 for oxygen in MgO, probably due to differences in the density of dislocations in the single crystals that were used (Van Orman and Crispin). We estimate bounds for 훿퐷퐺퐵 by using the largest (Oishi and Kingery) and smallest (Yang and Flynn) literature values for 퐷푣표푙; the smallest values are considered to represent intrinsic diffusion, and are in good agreement with theoretical calculations (Ita

-23 3 and Cohen). In this way, we obtain an experimental estimate 4x10 m /s < 훿퐷퐺퐵 <

2x10-21 m3/s, for oxygen diffusion at 1700 oC.

To compare our simulation results to experiment, we need to estimate the grain boundary width . We estimate  as the full-width at half maximum of the deviation in density associated with the grain boundary layer. These estimates for  are given in Table

-23 3 1, and fall in the range 6 – 10 Å. In this way, we obtain the value 훿퐷퐺퐵= 7x10 m /s for oxygen grain boundary diffusion at atmospheric pressure and 1700 oC, which falls in the range of experimental estimates described above.

4.5.2 Temperature dependence of grain boundary diffusivity

If the migration energy barrier, E, is independent of temperature, as it typically is for diffusion in crystals, then we have the familiar Arrhenius equation,

−퐸 퐷 = 퐷 푒푥푝 ( ). (4.6) 0 푅푇

Fitting our grain boundary diffusion data to Equation 4.6 yields apparent activation energies of 305 and 380 kJ/mol for Mg and O, respectively. These activation energies are surprisingly high. An experimental estimate for the activation energy for oxygen in grain boundary diffusion was 230 kJ/mol (Frost and Ashby), and experimental results for activation energies of Cr and Ni ion diffusion in MgO are 100-200 kJ/mol

(Wuensch and Vasilos, “Grain-Boundary Diffusion in MgO”; Osenbach and Stubican).

Furthermore, activation energies for atomic migration in MgO single crystals determined are in the range ~220-240 kJ/mol for Mg (Sempolinski and Kingery) and by simulations

(Ita and Cohen; Karki and Khanduja), and ~230-260 kJ/mol for O (not accounting for the formation energy of oxygen vacancies; (Ita and Cohen; Karki and Khanduja)). We would expect that the activation energy in a grain boundary would be smaller than in the

74 corresponding crystal interior, because atoms in crystal interior are fully surrounded by neighboring ions and thus will have larger migration barriers.

However, recent studies in metals has shown that grain boundaries can undergo structural transformations with changes in temperature (Suzuki and Mishin; Sellers et al.;

Frolov, Divinski, et al.; Frolov, Olmsted, et al.; Divinski et al.) that may contribute to the temperature dependence of grain boundary diffusion. In MgO, we suggest that the dynamics in grain boundaries at atmospheric pressure are similar to the dynamics of supercooled liquids. The reason for this suggestion is that, at zero pressure, the grain boundary diffusion coefficients extrapolate to the corresponding diffusion coefficients in the melt (Figure 4.6).

It is well known that in supercooled liquids, the system visits progressively deeper energy minima as the temperature decreases, and that this region of the energy landscape has higher barriers to diffusion (Sastry, Debenedetti, and Stillinger). Thus, as the temperature decreases, the dynamics of supercooled liquids slow down for two reasons.

First, there is a smaller probability of surmounting a given energy barrier, as described by the Arrhenius equation. But second, the energy barriers also increase in height.

Our results show that for MgO, with decreasing temperature the grain boundary becomes more ordered, with fewer undercoordinated ions, corresponding to energy minima with deeper minima (Figs 2-4). As in the case of supercooled liquids, the region of the energy landscape with deeper minima is also expected to have higher energy barriers for diffusion. If the migration energy barrier depends linearly on temperature,

퐸 = 퐸0 − 푐(푇 − 푇0) (4.7)

75 where c is a constant that describes the temperature dependence of the migration barrier, and E0 is the migration barrier at some reference temperature T0, then, combining (4.6) and (4.7):

푐 퐸 +푐푇 퐷 = [퐷 푒푥푝 ( )] 푒푥푝 (− 0 0). (4.8) 0 푅 푅푇

Equation 4.8 has exactly the same form as the Arrhenius equation, but its terms have different meanings. By comparing the numerator of the exponentials in Equations

4.6 and 4.8, we see that oxygen diffusion results described by a constant E = 380 kJ/mol are just as well described, for example, by an activation energy that varies from 220 kJ/mol at 1500 K to 175 kJ/mol at 2000 K (E0 = 200 kJ/mol, T0 = 2000 K, c = 0.09 kJ/K); this range of activation energies is in reasonable agreement with the expectations described in the first paragraph of this section.

4.5.3 Pressure dependence of grain boundary diffusivity

As discussed above, the dynamics in grain boundaries at atmospheric pressure are similar to those of supercooled liquids, as evidenced by the extrapolation of the grain boundary diffusion coefficients to the corresponding diffusion coefficients of the melt

(Figure 4.6). However, the situation is different at higher pressures – for example, at P =

8 GPa, the grain boundary diffusion coefficients do not extrapolate to the values in the melt, but instead extrapolate to values that are significantly smaller.

Thus, at higher pressure the grain boundary does not behave like a liquid. This change occurs because at higher pressure the structure becomes more ordered and more like the crystal, as shown in Figs. 2, 3 and 5. This change in structure leads to strongly

76 decreasing diffusion constants with increasing pressure; in fact, the rate of diffusion above ≈ 10 GPa is so low that the diffusive regime is not reached in the timescales of our simulations, even at the highest temperatures simulated.

The pressure dependence of the diffusion coefficient is often described by an activation volume 푉푎, as follows

휕 ln 퐷 −푉 ( ) = 푎 (4.9) 휕푃 푇 푅푇

Figure 4.7 shows results for the pressure dependence of grain boundary diffusion coefficients, with activation volumes obtained by fitting to the data at different temperatures. At low pressures, the activation volumes are approximately 10 cm3/mol, but they decrease with pressure and are approximately 5-7 cm3/mol at ~10 GPa, as shown in Table 4.2. This claim is further reinforced by simulations run at 2000 °C, as a second order polynomial fit for 푉푎, allowing 푉푎 to change as a function of pressure, better describes the data (as estimated by a nested F-test, p-values 0.02,0.04 for Mg and O respectively) than the linear model described in Equation 4.9. Table 4.3 shows the parameters of this fit.

Table 4.2: Activation volumes as a function of temperature and pressure. Uncertainties are 1σ standard errors.

Pressure Temperature Magnesium activation Oxygen activation

[GPa] [°C] volume [cm3/mol] volume [cm3/mol]

0-3 1800 7.91(16) 10.8(7)

0-8 2000 8.3(3) 11.7(4)

0-4 2000 10.4(2) 15.1(4)

6-12 2300 5.29(16) 7.8(3)

8-12 2600 6(7) 9(11)

Activation

volume dVa/dP

Species [cm^3/mol] [cm^3/mol/Pa] D0 F statistic P (F statistic)

Mg 10.0(4) -3.81(57)x10-10 1.57 (1.04-1.15)x10-9 44 0.022

O 14.2(8) -5.1(10)x10-10 2.38 (2.18-2.63)x10-10 22 0.042

Figure 4.7: Grain boundary diffusion coefficients as a function of pressure. Error bars represent the standard error of the estimate of the diffusion coefficient.

For comparison, experiments show that the activation volume for magnesium in the grain boundary is ~5 cm3/mol between 16-25 GPa (James A Van Orman et al.); the activation volume from our simulations are in good agreement with this value. Note that the activation volume for grain boundary diffusion is larger than that for diffusion in single crystal MgO, which is 2.7-3.1 cm3/mol over the pressure range 0-25 GPa (James A

Van Orman et al.).

This comparison shows that the decrease with pressure of diffusivity is stronger in grain boundaries than within crystal interiors; our simulations show that this stronger effect occurs due to the change in structure within the grain boundary with pressure, which increases order and inhibits diffusion. Further work is necessary to understand how the pressure dependence of grain boundary diffusion changes at pressures of the lower mantle (>23 GPa). It is likely that the activation volume continues to decrease with

79 pressure, as it becomes less compressible (and more ordered), as observed in simulations for Mg and O diffusion in single crystal MgO at pressures beyond ~25 GPa (Ita and

Cohen; Karki and Khanduja).

4.6 Conclusions

The grain boundary diffusion coefficients and activation volumes obtained here from molecular dynamics simulations are in reasonable agreement with experimental values, and provide additional insight on how diffusivity varies with pressure and temperature. For both Mg and O, the apparent activation energy for grain boundary diffusion is larger than the migration energy for volume diffusion in MgO. This strong temperature dependence is connected to a gradual shift to more ordered structures with decreasing temperature. This behavior is analogous to that of supercooled liquids at temperatures near the glass transition, where the energy barriers to atomic migration increase with decreasing temperature, and we show that the strong temperature dependence for grain boundary diffusion can be explained by a modest migration energy that is linearly dependent on temperature.

Grain boundary diffusion coefficients decrease sharply with increasing pressure, with activation volumes ~2-3 times larger than the migration volume for Mg and O diffusion within an MgO crystal at 0-4 GPa. The strong decrease in diffusivity with increasing pressure is again connected to an increase in atomic order, and an energy landscape with larger barriers to atomic migration, within the grain boundary under compression. As pressure increases beyond 4 GPa, the activation volume for grain

80 boundary diffusion decreases significantly, but remains larger than the corresponding migration volume in single-crystal MgO up to at least 12 GPa. At higher pressures, the excess volume of the grain boundary, and the fraction of undercoordinated atoms within it, continue to decrease. Whereas at low pressures the chemical transport properties of the grain boundary are similar to those of a supercooled liquid or glass, under high compression the grain boundary becomes more ordered, and its behavior becomes more similar to that of crystalline MgO.

Quantitative extrapolation of our results to the high pressures and temperatures of

Earth’s deep lower mantle is uncertain. However, based on the absence of any diffusion in a 100 ns simulation we can place an upper limit of 10-13 m2/s on the grain boundary diffusion coefficients at 135 GPa and 3500 oC, conditions relevant to Earth’s core-mantle boundary region. This result suggests that grain boundary transport is considerably less efficient at core-mantle boundary conditions than has been inferred from experimental results at lower pressures and temperatures (James A Van Orman et al.; Hayden and

Watson, “A Diffusion Mechanism for Core–Mantle Interaction”), with a maximum diffusion length scale for Mg and O diffusion on the order of 100 m.

5 An analysis of dominant transport mechanism as a function of

grain size

Diffusive transport in bulk systems of MgO occurs through multiple pathways, which scale differently. There are at least three routes for diffusion in pure crystalline materials: Triple junctions, grain boundaries, and volume diffusion. Triple junctions, which are regions where grain boundaries intersect, which have diffusivities possibly on the order of 103 faster than grain boundary diffusion, provide total fluxes that scale linearly with increasing grain size (as they are one-dimensional defects). This limits their contribution to the total flux, and they have not been considered here. Grain boundaries, the subject of Chapters 3 and 4Error! Reference source not found., provide total fluxes that scale as the square of the grain size (these are two dimensional defects). Volume diffusion, which we discuss in Chapter 2 and is driven by impurities in cationic MgO, provides total fluxes that scale as the cube of the grain size. The modified Hart equation

(Hart; Dohmen and Milke) can be used to obtain a grain size where grain boundary diffusion and volume diffusion contribute equally to bulk transport

퐷표푣푒푟푎푙푙 = 퐷퐺퐵푓푣 + (1 − 푓푣)퐷푣표푙 (5.1)

Where 퐷표푣푒푟푎푙푙 is an effective diffusion coefficient which reflects total bulk transport, 퐷퐺퐵 is grain boundary diffusion, 푓푣 is the volume fraction of the grain boundary

82 region, and 퐷푣표푙 is the volume diffusion coefficient. Taking the ratio of the volume and diffusion coefficients, we obtain

퐷 1−푓 퐺퐵 = 푣 (5.2) 퐷푣표푙 푓푣

As this is a scaling argument, we will ignore grain geometry, and assume a simple cubic grain. Thus we have

푑2훿 훿 푓 = ≈ (5.3) 푣 푑2(푑+훿) 푑

Where 훿 is the grain boundary width and 푑 is the grain size. Combining equations

5.2 and 5.3, we obtain the diameter where transport is equally facilitated by volume and grain boundary diffusion, 푑푐푟푖푡

퐷퐺퐵 푑푐푟푖푡 = 훿 (1 + ) (5.4) 퐷푣표푙

With the results in Table 4.1 and the relationship for volume diffusivity found in

Chapter 2, we can estimate 푑푐푟푖푡 for polycrystalline systems with a given impurity concentration. The results of such an estimation are shown in Figure 5.1, for an impurity concentration of 100 ppm. As the grain boundary diffusivity is much larger than the volume diffusivity for relevant concentrations, the value of 푑푐푟푖푡 is almost directly proportional to the impurity concentration within the cationic lattice. A more comprehensive analysis is possible which would give a more precise value of 푑푐푟푖푡, depending on the nature of impurities (as a function of their enthalpy of binding), but this is a scaling argument, and should be used as an order of magnitude estimation. From

Figure 5.1 we see that volume diffusion provides the highest contribution to mass transport in the upper mantle, where the grain sizes are likely on the order of millimeters

(Karato). The pressure dependence of diffusion at pressures above 25 GPa requires

83 extrapolation, but provided the trends remain, grain boundary diffusion in the core-mantle boundary region would not be a significant driver of mass transport, which would imply that bulk transport in systems deep in the mantle may be accurately modeled without reference to it.

Figure 5.1: grain sizes where lattice diffusion and grain boundary diffusion contribute equally to transport given a hypothetical defect cation with 100 ppm concentration.

6 Appendix

6.1 Algorithms/Scripts used (python code)

This section contains the most critical algorithms, presented as pieces of python code, which are relevant to the results presented in the text.

6.1.1 Identification of vacant site import pickle import MDAnalysis.analysis.distances from MDAnalysis.lib.distances import distance_array as mld from os.path import join as osp import itertools mgs = u2.select_atoms('not name O1') vps = [] vpsAlt={} tet = u2.select_atoms('name Mg2') count=20 eighteen=5832**(1/3) latConv = mgs.dimensions[0]/eighteen search=MDAnalysis.lib.NeighborSearch.AtomNeighborSearch(mgs,box=u2.dimensions) specialCases=[] sepdists = [] for ts in u2.trajectory: latSpots=(mgs.atoms.positions/latConv) remainder = np.mean(latSpots%1,axis=0) latIn=np.round(latSpots-remainder) latIn[latIn<0]+=18 jk=np.sum(latIn[0])%2 latIndices=np.sum(latIn*np.array([1e4,1e2,1]),axis=1) possibilities=np.sum(np.vstack(list(itertools.filterfalse(lambda x: sum(x)%2!=jk,itertools.product(range(18),range(18),range(18)))))*np.array([1e4,1e2,1]), axis=1) trialPos = np.setdiff1d(possibilities,latIndices) if len(trialPos)==1: vacPos=(trialPos*np.array([1e-4,1e-2,1])%100//1+remainder)*latConv vps.append(vacPos) else: specialCases.append(ts.frame) class vpi: def __init__(self,x): self.positions=x

def __str__(self): return self.positions.__str__() def __repr__(self): return self.positions.__repr__() tocheck = vpi(((np.array([1e-4,1e-2,1])[:,np.newaxis] * np.setdiff1d(possibilities,latIndices)).T % 100//1+remainder) *latConv) searchout=search.search(tocheck,2.5) #interstitial bound to a vacancy if len(searchout)==1 and np.sum(latIn[searchout.indices[0]])%2!=jk: vps.append(searchout.positions[0]) elif len(searchout)==0:#Multiple REAL vacancies vacPos=(trialPos[0]*np.array([1e-4,1e-2,1])%100//1+remainder)*latConv vps.append(vacPos) vpsAlt[ts.frame]=((np.array([1e-4,1e- 2,1])[:,np.newaxis]*np.setdiff1d(possibilities,latIndices)).T%100//1+remainder)*latConv else: allvposs=[] for i in tocheck.positions: searchout=search.search(vpi(i),2.5) if not len(searchout): allvposs.append(i) counter=0 imax=-1 distmax=-1 if len(allvposs)==0: for i in tocheck.positions: searchout=search.search(vpi(i),2.5) searchdist =np.min(MDAnalysis.analysis.distances.distance_array( searchout.positions, vpi(i).positions,u2.dimensions)) if distmax < searchdist: imax=counter distmax=searchdist counter+=1 allvposs.append(tocheck.positions[imax]) vps.append(allvposs[0]) vpsAlt[ts.frame]=allvposs progressBar.value+=1 sepdists.append(mld(tet.atoms.positions,vps[-1],box=u2.dimensions)[0,0]) np.save(os.path.join(folder,'pickleout_vactrackcat{:d}.npy'.format(McCounter)), np.vstack(vps))

6.1.2 Grain boundary lattice creation import re import itertools import numpy as np def build_lattice_double(a=0.2106, theta=0, b=None, c=None, n=13): """Build a rotated lattice and stack a twinned grain boundary rotation is done in the xz plane (around y axis)

For k₁, k₂ in the naturals, here are rules about n: nz = (2k₁+1) nx = k₂*tan(θ), (nx+k₂) % 2 == 1

nx, ny, nz must be integers a*(nx,ny,nz) must be greater than 0.55*8=4.4 nm """

theta %= np.pi/4 nx = np.min((n*2,50)) ny = np.min((n,50)) nz = (n+8)//4*2+1 nk = 0 thetaDiff = 4 if theta != 0: for k in range(1, nx+1): for j in range(k+1): thetaTest = np.arctan(j/k) if abs(thetaTest-theta) < thetaDiff and (k+j) % 2 == 0: thetaDiff = abs(thetaTest-theta) nk = k nj = j print(nj,nk) theta = np.arctan(nj/nk) nxOld = nx nx = np.sqrt(nj**2+nk**2) else: nxOld = nx # if a*nx<4.5 or a*ny<4.5 or a*nz<4.5: # amount = int(np.ceil(4.4/min(nx,ny,nz))) # nx = nx*amount # ny = ny*amount # nz = nz*amount nz = nz*np.cos(theta) mult = nxOld//nx if mult:

nx *= mult thet = -theta Q = np.array([[np.cos(thet), -np.sin(thet)], [np.sin(thet), np.cos(thet)]]) if isinstance(b, type(None)): b = a if isinstance(c, type(None)): c = b l = 0 xyz = [] xz = [] indxz = [] molType = [] ia = 0 ic = 0 while True: ind = np.array([ia, ic]) abc = np.dot(Q, ind)*a proposed = abc if proposed[0] >= a*nx+1e-12 or proposed[1] < 0: ind[1] += 1 while(proposed[0] >= 0 and proposed[1] < a*nz): ind[0] -= 1 proposed = np.dot(Q, ind)*a ind[0] += 1 proposed = np.dot(Q, ind)*a if proposed[1] >= a*nz or proposed[0] > a*nx+1e-12: break xz.append(np.dot(Q, ind)*a) ind[0] += 1

l += 1 ia = ind[0] ic = ind[1] indxz.append([ia, ic]) print(a) xyz = np.zeros((0, 3)) indxz = np.array(indxz) xz = np.array(xz) xn, xm = np.shape(xz) selection = np.where([np.min(((xz[aIndex, 0]+a*nx)-xz[:, 0])**2 + (xz[aIndex, 1]-xz[:, 1])**2) < a/120 for aIndex in range(xn)]) xz = np.delete(xz, selection, axis=0) indxz = np.delete(indxz, selection, axis=0) xz[0, xz[0, :] > a*nx] -= a*nx

88 for j in range(0, ny//2*2): xyz = np.vstack((xyz, np.array([xz[:, 0], np.ones(np.shape(xz[:, 0]))*j*a, xz[:, 1]]).T)) molTypej = (np.sum(indxz, axis=1)+j) % 2 molType = np.hstack((molType, molTypej)) nx2, nDimensions = np.shape(xyz) number = np.arange(0, 4*nx2, 1) xyz[:, 2] += a/2 if theta != 0: nz = nz + np.cos(theta) box = np.array([nx*a, ny//2*2*a, 4*nz*a]) print(xyz.shape) xyz2 = xyz.copy() xyz2[:, 2] = 2*nz*a - xyz2[:, 2] xyz = np.vstack((xyz, xyz2)) print(xyz.shape) xyz2 = xyz.copy() xyz2[:, 2] = 2*nz*a + xyz2[:, 2] xyz = np.vstack((xyz, xyz2)) print(xyz.shape) molType2 = molType.copy() molType2[molType2 == 1] = 3 molType2[molType2 == 0] = 1 molType2[molType2 == 3] = 0 molType = np.hstack((molType, molType2)) molType2 = molType.copy() molType = np.hstack((molType, molType2)) components = np.array([number, molType]) indices = np.argsort(components[1, :]) xyz = xyz.T xyz = xyz[:, indices] components = components[:, indices] components[0, :] = components[0, :]+1 return xyz, components, box, theta

6.1.3 Grain boundary width determination

#need the output .gro file from an energy minimization from gromacs to identify atomic positions import scipy.signal as scsig import numpy as np import MDAnalysis toFit = [(int(x.split(' ')[0]),int(x.split(' ')[1])) for x in """1800 0 2300 10 2000 50""".split('\n')] number=1 maxBins=172 binstep=2.5 stepMax=11 #for stepMax in [2,6,11,18]: # for binstep in [1,1.5,2,2.5,3,4]: binsteps = [2.5] gbwidth=2 checkBins = np.arange(0,172,1.5) xvals = [] yvals = [] gbwidths={} if True: bins=np.arange(0,172,binstep) for tempture,press in toFit: die=True poszsMg = [] poszsO = [] for number in range(1,stepMax): if not die: die=True with MDAnalysis.Universe(minimized_grofile[number]) as uni: mgs = uni.select_atoms(‘name Mg1’) oxys = uni.select_atoms(‘name O1’) poszsMg.append(np.hstack([ mgs.atoms.positions[:,2].copy() for ts in uni.trajectory]) poszsO.append(np.hstack([ mgs.atoms.positions[:,2].copy() for ts in uni.trajectory]) maxBins=max(maxBins,uni.trajectory.dimensions[2]) die=False # cutoff=2.13 # counter = 0 if not die: silencesMg=(np.histogram(poszsMg,bins=bins,density=True)) silencesO=( np.histogram(poszsO,bins=bins,density=True)) xysMg=(np.vstack([((silencesMg[1][:-1]+silencesMg[1][1:])/2),silencesMg[0]]))

xysO=( np.vstack([((silencesO[1][:-1] +silencesO[1][1:]) /2),silencesO[0]])) densities= (xysMg[1,:]*0.024305+xysO[1,:]*0.015999)*67116/scc.Avogadro/(dims[0]*dims[1]*scc .angstrom**3) xvals.append(xysMg[0,:]) yvals.append(densities)

convolver = scsig.cosine(int(gbwidth*2/(binstep/10))) convolver/=np.sum(convolver) minima = [] counter=0 scales=[0.9,0.9,0.975] centys =[] centxs = np.arange(-20,22.5,2.5) for x,y in zip(xvals,yvals): y2=y[y>2000] x2=x[y>2000] mins = scsig.argrelextrema(y2,np.less)[0] mins = mins[y2[mins]0])*scales[counter]] centy=[] for mi in mins: if mi-8>0 and mi+90: centy.append([*y2[mi-8:],*y2[0:mi+9-len(y)]]) centys.append(centy) for centy in centys: centymean = np.mean(np.array(centy),axis=0) meanc=np.mean(centymean[centymean!=np.min(centymean)]) getfwhm=(centymean-meanc)-(np.min(centymean)-meanc)/2 spline = scint.UnivariateSpline(centxs,getfwhm,s=0) gbw=np.diff(spline.roots()) gbwidths[(tempture,press)]=gbw

6.1.4 Grain boundary atom assignment and grain assignment import MDAnalysis import time import shlex, subprocess import multiprocessing import matplotlib import matplotlib.patches as mpatches import matplotlib.animation as animation import matplotlib.path as path import matplotlib.pyplot as plt import numpy as np import scipy.constants as scc import csv import os import scipy.cluster class Lattice(MDAnalysis.Universe): """ Lattice Class: a subclass of MDAnalysis.Universe with built-in nearest neighbor searching functions - See GetLattice PlotLattice PlotNN (Plot Nearest Neighbors) """ def __init__(self, *args, **kwargs): super().__init__(*args,**kwargs)

def GetLattice(self,frame=None,selection='name Mg1',refinedSelection='all',reference='all',fromResname=False,NNearest=6): """GetLattice selection: If fromResname is true, this is a list of integer resids, otherwise a SQL style selection query - The atoms that should be the center of the nearest neighbor search refinedSelection: If selection needs an AND in the SQL syntax, just fill this with what should be on the other - side of the AND, further refining the selection reference: Atoms that should be used to compare against fromResname: Bool, if true, selection should be an np array or list of integers with values corresponding to the - resids of the atoms that should be in the selection NNearest: How many atoms to include in the returned results, 6 or 15 make sense. """ if fromResname: Mgs = self.residues[selection-1].atoms else:

Mgs = self.select_atoms(selection) Mgs = Mgs.select_atoms(refinedSelection) self.Mgs = Mgs MgRefs = self.select_atoms('name Mg1') ORefs = self.select_atoms('name O1') Os = self.select_atoms(reference) self.Os = Os neighborSearch = MDAnalysis.lib.NeighborSearch.AtomNeighborSearch(Os,box=self.dimensions) atomList = [] for i in Mgs: ag = neighborSearch.search(i,6.5) dists = MDAnalysis.lib.distances.distance_array(ag.atoms.positions,np.array([i.position]),self.di mensions) #TODO: save which atoms are in the grain boundary. atomList.append(sorted([x for x in dists.flatten() if x > 0])) distances = np.array([x[:NNearest] for x in atomList]) myDists = distances totalmyDists = distances.copy() totalinds = Mgs.atoms.resids.copy().astype(int) self.myDists = totalmyDists self.myinds = totalinds return myDists, totalinds

def GetNNInfo(self,frame=None,selection='all',reference='all',NNearest=7,namingScheme='a ll',folder='/home/ke7kto/Documents/Research/gromacs/analysis'): """Selection: The atoms to compute the nearest neighbors for Reference: The atoms to compute the nearest neighbors with NNearest: The number of nearest neighbors to look for NamingSchema: A unique way to name the output """ frame = frame or self.trajectory.frame toGetNNs = self.select_atoms(selection) connectionGrid = np.zeros((toGetNNs.n_atoms,6),dtype=np.int32) connectionGrid[:,0]=toGetNNs.atoms.resids atomDists = np.zeros((toGetNNs.n_atoms,6),dtype=np.float64) atomRef = self.select_atoms(reference) MgRefs = self.select_atoms('name Mg1') ORefs = self.select_atoms('name O1') neighborSearch = MDAnalysis.lib.NeighborSearch.AtomNeighborSearch(atomRef,box=self.dimensions) MgneighborSearch = MDAnalysis.lib.NeighborSearch.AtomNeighborSearch(ORefs,box=self.dimensions)

OneighborSearch = MDAnalysis.lib.NeighborSearch.AtomNeighborSearch(MgRefs,box=self.dimensions) for i in toGetNNs: if i.name == 'Mg1': ag = MgneighborSearch.search(i,8) elif i.name == 'O1': ag = OneighborSearch.search(i,8) else: print("FAILURE at time "+str(frame)) dists = MDAnalysis.lib.distances.distance_array(ag.atoms.positions,np.array([i.position]),self.di mensions).flatten() closests = np.argsort(dists) connectionGrid[i.resid-1]=ag.resids[closests[:6]] atomDists[i.resid-1]=dists[closests[:6]] self.connectionGrid=connectionGrid self.connectDists=atomDists np.savetxt(os.path.join(folder,"connectionGrid_{:s}_{:d}.csv".format(namingScheme,fra me)),connectionGrid) np.savetxt(os.path.join(folder,"connectionDists_{:s}_{:d}.csv".format(namingScheme,fra me)),atomDists)

def PlotLattice(self,dists=None,function=np.sum,xlabel='',xlims=None,title='',ax=None): """Plots a histogram of distances to the nearest neighbors.""" dists = dists or self.myDists distances = function(dists) if type(ax) is type(None): fig, ax = plt.subplots() histInfo = ax.hist(distances,bins=500) ax.set_xlabel(xlabel+' (\\AA)') ax.set_ylabel('Mg atom count') ax.set_xlim(xlims) ax.set_title(title) return ax

def PlotNN(self,distsSet=None,labels=['Grain Boundary','Bulk'],figsize=(8,8)): """Plots the nearest neighbors, which are either calculated as part of the function call, or supplied as the distsSet parameter """ if type(distsSet) is type(np.ndarray):

pass else: distsSet = distsSet or np.array([self.GetLattice(0,'name Mg1','prop z > 39 and prop z < 47')[0],self.GetLattice(0,'name Mg1','prop z < 39 and prop z > 10')[0]]) fig, axes = plt.subplots(nrows=distsSet.shape[1]+1,ncols=len(distsSet),figsize=figsize,sharex='col') distMeans = np.mean(np.vstack(distsSet),axis=1) for i in range(len(distsSet)): axes[0][i].hist(distMeans,bins=50) axes[-1][i].set_xlabel(labels[i]) axes[0][0].set_ylabel('mean') for i in range(1,distsSet.shape[1]+1): for j in range(len(distsSet)): axes[i][j].hist(distsSet[j][:,i-1],bins=50) if i==1: axes[i][0].set_ylabel('{:d}st'.format(i)) elif i==2: axes[i][0].set_ylabel('{:d}nd'.format(i)) elif i==3: axes[i][0].set_ylabel('{:d}rd'.format(i)) else: axes[i][0].set_ylabel('{:d}th'.format(i))

plt.tight_layout() return fig, axes, distsSet

def CutoffSelection(self,cutoff=None,selection='name Mg1',refinedSelection='all',reference='name O1',fromResname=False,frame=0,NNearest=6): self.GetLattice(selection=selection, refinedSelection=refinedSelection, reference=reference, fromResname=fromResname, frame=frame, NNearest=NNearest) cutoff = cutoff or np.mean(self.myDists)*(1+np.sqrt(3))/2 inds = self.myinds[np.where(self.myDists[:,-1]>cutoff)[0]] return inds

7 Works-Cited List

Aleshin, A. N., et al. “Evidence of Structure Transformation in Σ = 5 Near-Coincidence

Grain Boundaries.” Scripta Metallurgica, vol. 19, no. 10, 1985, pp. 1135–40,

doi:10.1016/0036-9748(85)90223-6.

Ammann, M. W., et al. “Diffusion of Aluminium in MgO from First Principles.” Phys.

Chem Minerals, vol. 39, no. 6, June 2012, pp. 503–14, doi:10.1007/s00269-012-

0506-z.

Barnhoorn, Auke, et al. “Transition from Elastic to Inelastic Deformation Identified by a

Change in Trend of Seismic Attenuation, Not Seismic Velocity-A Laboratory Study.”

Geophysical Research Abstracts, vol. 18, 2016, pp. 2016–5937.

Bassani, F., and F. G. Fumi. “The Interaction between Equilibrium Defects in the Alkali

Halides: The ‘Ground State’ Binding Energies between Divalent Impurities and

Vacancies.” Il Nuovo Cimento, vol. 11, no. 3, Mar. 1954, pp. 274–84.

Bean, Jonathan J., et al. “Atomic Structure and Electronic Properties of MgO Grain

Boundaries in Tunnelling Magnetoresistive Devices.” Scientific Reports, vol. 7,

Nature Publishing Group, Apr. 2017, p. 45594, doi:10.1038/srep45594.

Berendsen, H.J.C., et al. “GROMACS: A Message-Passing Parallel Molecular Dynamics

Implementation.” Computer Physics Communications, vol. 91, no. 1–3, Sept. 1995,

pp. 43–56, doi:10.1016/0010-4655(95)00042-E.

Berendsen, H J C, et al. “Molecular Dynamics with Coupling to an External Bath.” The

Journal of Chemical Physics, vol. 81, no. 8, Oct. 1984, pp. 3684–90,

doi:10.1063/1.448118.

Carroll, J. C. G., et al. “Theoretical Study of the Defect Distribution of Trivalent Cation

Impurities in MgO.” Journal of Materials Science, vol. 23, 1988, pp. 2824–36.

Crispin, Katherine L., and James A. Van Orman. “Influence of the Crystal Field Effect on

Chemical Transport in Earth’s Mantle: Cr3+ and Ga3+ Diffusion in Periclase.”

Physics of the Earth and Planetary Interiors, vol. 180, no. 3–4, June 2010, pp. 159–

71, doi:10.1016/j.pepi.2009.12.004.

Darden, Tom, et al. “Particle Mesh Ewald: An N Log( N ) Method for Ewald Sums in

Large Systems.” The Journal of Chemical Physics, vol. 98, no. 12, June 1993, pp.

10089–92, doi:10.1063/1.464397.

De Vita, A., M. J. Gillan, et al. “Defect Energetics in MgO Treated by First-Principles

Methods.” Physical Review B, vol. 46, no. 20, Nov. 1992, pp. 12964–73,

doi:10.1103/PhysRevB.46.12964.

De Vita, A., M J Gillan, et al. “Defect Energetics in Oxide Materials from First

Principles.” Phys. Rev Lett., vol. 68, June 1992, pp. 3319–22.

Debenedetti, Pablo G., and Frank H. Stillinger. “Supercooled Liquids and the Glass

Transition.” Nature, vol. 410, Nature Publishing Group, Mar. 2001, p. 259,

http://dx.doi.org/10.1038/35065704.

Department of Energy. “CRA-2014 Appendix MgO: Magnesium Oxide as an Engineered

Barrier.” Appendix MgO: Magnesium Oxide as an Engineered Barrier, United

States Department of Energy, Mar. 2014,

http://www.wipp.energy.gov/library/cra/CRA-

2014/CRA/Appendix_MgO/Appendix_MgO.htm.

Divinski, Sergiy V, et al. “Diffusion and Segregation of Silver in Copper Σ 5(310) Grain

Boundary.” Physical Review B, vol. 85, no. 14, Apr. 2012, p. 144104,

doi:10.1103/PhysRevB.85.144104.

Dohmen, R., and R. Milke. “Diffusion in Polycrystalline Materials: Grain Boundaries,

Mathematical Models, and Experimental Data.” Reviews in Mineralogy and

Geochemistry, vol. 72, no. 1, GeoScienceWorld, Jan. 2010, pp. 921–70,

doi:10.2138/rmg.2010.72.21.

Dreyfus, R. W., and R. B. Laibowitz. “Anelastic and Dielectric Relaxation Due to

Impurity-Vacancy Complexes in NaCl Crystals.” Physical Review, vol. 135, no. 5A,

Aug. 1964, pp. A1413–22.

Dziewonski, Adam M., and Don L. Anderson. “Preliminary Reference Earth Model.”

Physics of the Earth and Planetary Interiors, vol. 25, no. 4, 1981, pp. 297–356,

doi:https://doi.org/10.1016/0031-9201(81)90046-7.

Eiler, John M., et al. “Intercrystalline Stable Isotope Diffusion: A Fast Grain Boundary

Model.” Contributions to Mineralogy and Petrology, vol. 112, no. 4, Dec. 1992, pp.

543–57, doi:10.1007/BF00310783.

Einstein, Albert. Investigations on the Theory of the Brownian Movement. Edited by R.

Furth, Dover, 1956,

https://books.google.com/books?hl=en&lr=&id=X5iRDQAAQBAJ&oi=fnd&pg=P

A139&dq=INVESTIGATIONS+ON+THE+THEORY+OF+THE+BROWNIAN+M

OVEMENT+Einstein&ots=-VW0mfaAvg&sig=-

b8s2tYZsskwzMdlmYCHYDS3SFs.

Fisher, J. C. “Calculation of Diffusion Penetration Curves for Surface and Grain

Boundary Diffusion.” Journal of Applied Physics, vol. 22, no. 1, {AIP} Publishing,

Jan. 1951, pp. 74–77, doi:10.1063/1.1699825.

Frolov, Timofey, S. V Divinski, et al. “Effect of Interface Phase Transformations on

Diffusion and Segregation in High-Angle Grain Boundaries.” Physical Review

Letters, vol. 110, no. 25, June 2013, p. 255502,

doi:10.1103/PhysRevLett.110.255502.

Frolov, Timofey, David L. Olmsted, et al. “Structural Phase Transformations in Metallic

Grain Boundaries.” Nature Communications, vol. 4, no. 1, Oct. 2013, p. 1899,

doi:10.1038/ncomms2919.

Frost, Harold J., and Michael F. Ashby. Deformation Mechanism Maps: The Plasticity

and Creep of Metals and Ceramics. Pergamon Press, 1982.

Garnero, Edward J., et al. Ultralow Velocity Zone at the Core-Mantle Boundary.

American Geophysical Union (AGU), 2011, pp. 319–34, doi:10.1029/gd028p0319.

Gourdin, W. H., and W. D. Kingery. “The Defect Structure of MgO Containing Trivalent

Cation Solutes: Shell Model Calculations.” Journal of Materials Science, vol. 14,

no. 9, Kluwer Academic Publishers, Sept. 1979, pp. 2053–73,

doi:10.1007/BF00688410.

Harding, J. H., and D. J. Harris. “Simulation of Grain-Boundary Diffusion in Ceramics

by Kinetic Monte Carlo.” Physical Review B, vol. 63, no. 9, Jan. 2001, p. 094102,

doi:10.1103/PhysRevB.63.094102.

Hart, E. W. “On the Role of Dislocations in Bulk Diffusion.” Acta Metallurgica, vol. 5,

no. 10, Pergamon, Oct. 1957, p. 597, doi:10.1016/0001-6160(57)90127-X.

Hayden, Leslie A., and E. Bruce Watson. “A Diffusion Mechanism for Core–Mantle

Interaction.” Nature, vol. 450, no. 7170, Springer Nature, Nov. 2007, pp. 709–11,

doi:10.1038/nature06380.

---. “Grain Boundary Mobility of Carbon in Earth’s Mantle: A Possible Carbon Flux from

the Core.” Proceedings of the National Academy of Sciences, vol. 105, no. 25,

National Academy of Sciences, 2008, pp. 8537–41, doi:10.1073/pnas.0710806105.

Hirata, K., et al. “High Temperature Thermal Expansion of ThO2, MgO and Y2O3 by X-

Ray Diffraction.” Journal of Materials Science, vol. 12, no. 4, 1977, pp. 838–39.

Hirth, Greg, and David L. Kohlstedt. “Experimental Constraints on the Dynamics of the

Partially Molten Upper Mantle: Deformation in the Diffusion Creep Regime.”

Journal of Geophysical Research: Solid Earth, vol. 100, no. B2, Feb. 1995, pp.

1981–2001, doi:10.1029/94JB02128.

Hoover, William G. “Canonical Dynamics: Equilibrium Phase-Space Distributions.”

Physical Review A, vol. 31, no. 3, American Physical Society, Mar. 1985, pp. 1695–

97, doi:10.1103/PhysRevA.31.1695.

Hoshino, T., et al. “Local-Density-Functional Calculations for Defect Interactions in Al.”

Physical Review B - Condensed Matter and Materials Physics, vol. 53, no. 14,

American Physical Society, Apr. 1996, pp. 8971–74,

doi:10.1103/PhysRevB.53.8971.

Humphrey, William, et al. “VMD -- Visual Molecular Dynamics.” Journal of Molecular

Graphics, vol. 14, 1996, pp. 33–38.

Inbar, Iris, and R. E. Cohen. “High Pressure Effects on Thermal Properties of MgO.”

Geophysical Research Letters, vol. 22, no. 12, Wiley Online Library, 1995, pp.

1533–36, doi:10.1029/95GL01086.

Inderjeet Kaur., et al. Fundamentals of Grain and Interphase Boundary Diffusion. John

Wiley, 1995.

100

Ita, Joel, and Ronald E. Cohen. “Diffusion in MgO at High Pressure: Implications for

Lower Mantle Rheology.” Geophysical Research Letters, vol. 25, no. 7, American

Physical Society ({APS}), Apr. 1998, pp. 1095–98, doi:10.1029/98GL50564.

Janssens, Koenraad G. F., et al. “Computing the Mobility of Grain Boundaries.” Nature

Materials, vol. 5, no. 2, Feb. 2006, pp. 124–27, doi:10.1038/nmat1559.

Joesten, Raymond. “Grain-Boundary Diffusion Kinetics in Silicate and Oxide Minerals.”

Diffusion, Atomic Ordering, and Mass Transport: Selected Topics in Geochemistry,

edited by J Ganguly, Springer US, 1991, pp. 345–95, doi:10.1007/978-1-4613-9019-

0_11.

Jollands, M. C., et al. “Substitution and Diffusion of Cr2+ and Cr3+ in Synthetic

Forsterite and Natural Olivine at 1200–1500 °C and 1 bar.” Geochimica et

Cosmochimica Acta, vol. 220, Elsevier Ltd, Jan. 2018, pp. 407–28,

doi:10.1016/j.gca.2017.09.030.

Karakasidis, Theodoros E., and Madeleine Meyer. “Molecular Dynamics Simulation of

the Atomic Structure of a NiO Tilt Grain Boundary at High Temperature.” Modelling

and Simulation in Materials Science and Engineering, vol. 8, no. 2, IOP Publishing,

2000, p. 117, doi:10.1088/0965-0393/8/2/303.

Karato, Shun-ichiro, and H. A. Spetzler. “Defect Microdynamics in Minerals and Solid-

State Mechanisms of Seismic Wave Attenuation and Velocity Dispersion in the

Mantle.” Reviews of Geophysics, vol. 28, no. 4, 1990, p. 399,

doi:10.1029/RG028i004p00399.

Karato, Shun-ichiro, and Patrick Wu. “Rheology of the Upper Mantle: A Synthesis.”

Science, vol. 260, no. 5109, May 1993, pp. 771–78,

101

doi:10.1126/science.260.5109.771.

Karato, Shun Ichiro. “Grain-Size Distribution and Rheology of the Upper Mantle.”

Tectonophysics, vol. 104, no. 1–2, Elsevier, Apr. 1984, pp. 155–76,

doi:10.1016/0040-1951(84)90108-2.

Karki, B. B., et al. “Structure and Elasticity of MgO at High Pressure.” American

Mineralogist, vol. 82, no. 1–2, Feb. 1997, pp. 51–60, doi:10.2138/am-1997-1-207.

Karki, B. B., and G. Khanduja. “Vacancy Defects in MgO at High Pressure.” American

Mineralogist, vol. 91, no. 4, GeoScienceWorld, Apr. 2006, pp. 511–16,

doi:10.2138/am.2006.1998.

Katsura, Tomoo, et al. “Adiabatic Temperature Profile in the Mantle.” Physics of the

Earth and Planetary Interiors, vol. 183, no. 1–2, Elsevier, Nov. 2010, pp. 212–18,

doi:10.1016/j.pepi.2010.07.001.

Kim, Rakhwan, et al. “Improvement of Secondary Electron Emission Property of MgO

Protective Layer for an Alternating Current Plasma Display Panel by Addition of

TiO2.” Thin Solid Films, vol. 376, no. 1–2, Elsevier Sequoia SA, Nov. 2000, pp.

183–87, doi:10.1016/S0040-6090(00)01184-6.

Klemradt, U., et al. “Vacancy-Solute Interactions in Cu, Ni, Ag, and Pd.” PHYSICAL

REVIEW B, vol. 43, no. 12, Apr. 1990, pp. 15–1991.

Lacks, Daniel J, Gaurav Goel, et al. “Isotope Fractionation by Thermal Diffusion in

Silicate Melts.” Physical Review Letters, vol. 108, no. 6, Feb. 2012, p. 65901.

Lacks, Daniel J., et al. “Isotope Fractionation in Silicate Melts.” Nature, vol. 482, no.

7385, Feb. 2012, pp. E1–E1, doi:10.1038/nature10764.

Lacks, Daniel J, David B. Rear, et al. “Molecular Dynamics Investigation of Viscosity,

102

Chemical Diffusivities and Partial Molar Volumes of Liquids along the MgO–SiO2

Join as Functions of Pressure.” Geochimica et Cosmochimica Acta, vol. 71, no. 5,

Elsevier BV, Mar. 2007, pp. 1312–23, doi:10.1016/j.gca.2006.11.030.

Lacks, Daniel J., and Mark J. Osborne. “Energy Landscape Picture of Overaging and

Rejuvenation in a Sheared Glass.” Physical Review Letters, vol. 93, no. 25,

American Physical Society, Dec. 2004, p. 255501,

doi:10.1103/PhysRevLett.93.255501.

Landuzzi, Fabio, et al. “Molecular Dynamics of Ionic Self-Diffusion at an MgO Grain

Boundary.” Journal of Materials Science, vol. 50, no. 6, SPRINGER, Mar. 2015, pp.

2502–09, doi:10.1007/s10853-014-8808-9.

Le Claire, A. D. “The Analysis of Grain Boundary Diffusion Measurements.” British

Journal of Applied Physics, vol. 14, no. 6, IOP Publishing, June 1963, pp. 351–56,

doi:10.1088/0508-3443/14/6/317.

Lejček, Pavel. “Grain Boundaries: Description, Structure and Thermodynamics.” Grain

Boundary Segregation in Metals, Springer Berlin Heidelberg, 2010, pp. 5–24,

doi:10.1007/978-3-642-12505-8_2.

Lidiard, A. B. “XCII. Impurity Diffusion in Polar Crystals.” The London, Edinburgh, and

Dublin Philosophical Magazine and Journal of Science, vol. 46, no. 379, Informa

UK Limited, Aug. 1955, pp. 815–23, doi:10.1080/14786440808561233.

Lin, Jung-Fu, et al. “Deformation of Lower-Mantle Ferropericlase (Mg,Fe)O across the

Electronic Spin Transition.” Physics and Chemistry of Minerals, vol. 36, no. 10,

Springer Nature, Dec. 2009, pp. 585–92, doi:10.1007/s00269-009-0303-5.

Liu, W. L., et al. “Thermal Conduction in Nanocrystalline Diamond Films: Effects of the

103

Grain Boundary Scattering and Nitrogen Doping.” Applied Physics Letters, vol. 89,

no. 17, American Institute of Physics, Oct. 2006, p. 171915, doi:10.1063/1.2364130.

Løvvik, O. M., et al. “Prediction of Solute Diffusivity in Al Assisted by First-Principles

Molecular Dynamics.” Journal of Physics: Condensed Matter J. Phys.: Condens.

Matter, vol. 26, Dec. 2013, p. 10, doi:10.1088/0953-8984/26/2/025403.

Ma, Qing, et al. “Diffusion along [001] Tilt Boundaries in the Au/Ag System—II.

Atomistic Modeling and Interpretation.” Acta Metallurgica et Materialia, vol. 41,

no. 1, Jan. 1993, pp. 143–51, doi:10.1016/0956-7151(93)90346-T.

Mantisi, Boris, et al. “Structure and Transport at Grain Boundaries in Polycrystalline

Olivine: An Atomic-Scale Perspective.” Geochimica et Cosmochimica Acta, vol.

219, no. 1, Elsevier {BV}, Jan. 2017, pp. 160–76, doi:10.1016/j.gca.2017.09.026.

Marquardt, Katharina, Quentin M. Ramasse, et al. “Diffusion in Yttrium Aluminium

Garnet at the Nanometer-Scale: Insight into the Effective Grain Boundary Width.”

American Mineralogist, vol. 96, no. 10, MINERALOGICAL SOC AMER, Oct.

2011, pp. 1521–29, doi:10.2138/am.2011.3625.

Marquardt, Katharina, Gregory S. Rohrer, et al. “The Most Frequent Interfaces in Olivine

Aggregates: The GBCD and Its Importance for Grain Boundary Related Processes.”

Contributions to Mineralogy and Petrology, vol. 170, no. 4, Springer Berlin

Heidelberg, Oct. 2015, p. 40, doi:10.1007/s00410-015-1193-9.

McKenna, K. P., and A. L. Shluger. “First-Principles Calculations of Defects near a Grain

Boundary in {MgO}.” Physical Review B, vol. 79, no. 22, American Physical

Society ({APS}), June 2009, doi:10.1103/physrevb.79.224116.

McKenzie, D. R., et al. “Oxygen Grain-Boundary Diffusion in MgO.” Journal of the

104

American Ceramic Society, vol. 54, no. 4, Wiley, Apr. 1971, pp. 188–90,

doi:10.1111/j.1151-2916.1971.tb12261.x.

Michaud-Agrawal, Naveen, et al. “MDAnalysis: A Toolkit for the Analysis of Molecular

Dynamics Simulations.” Journal of Computational Chemistry, vol. 32, no. 10, Wiley

Subscription Services, Inc., A Wiley Company, July 2011, pp. 2319–27,

doi:10.1002/jcc.21787.

Mohammadzadeh, Roghayeh, and Mina Mohammadzadeh. “Effect of Grain Boundary

Misorientation on the Apparent Diffusivity in Nanocrystalline Aluminum by

Atomistic Simulation Study.” Journal of Applied Physics, vol. 124, no. 3, {AIP}

Publishing, July 2018, p. 035102, doi:10.1063/1.5033860.

Nosé, Shuichi. “A Unified Formulation of the Constant Temperature Molecular

Dynamics Methods.” The Journal of Chemical Physics, vol. 81, no. 1, July 1984, pp.

511–19, doi:10.1063/1.447334.

Ohnuma, Toshiharu, et al. “First-Principles Calculations of Vacancy-Solute Element

Interactions in Body-Centered Cubic Iron.” Acta Materialia, vol. 57, no. 20,

Pergamon, Dec. 2009, pp. 5947–55, doi:10.1016/j.actamat.2009.08.020.

Oishi, Y., and W. D. Kingery. “Oxygen Diffusion in Periclase Crystals.” The Journal of

Chemical Physics, vol. 33, no. 3, {AIP} Publishing, Sept. 1960, pp. 905–06,

doi:10.1063/1.1731286.

Osenbach, John W., and Vladimir S. Stubican. “Grain-Boundary Diffusion of 51Cr in

MgO and Cr-Doped MgO.” Journal of the American Ceramic Society, vol. 66, no. 3,

Wiley, Mar. 1983, pp. 191–95, doi:10.1111/j.1151-2916.1983.tb10015.x.

Perkins, Richard A., and Robert A. Rapp. “The Concentration-Dependent Diffusion of

105

Chromium in Nickel Oxide.” Metallurgical Transactions, vol. 4, 1973, pp. 193–205.

Reali, R., et al. “Modeling Viscosity of (Mg,Fe)O at Lowermost Mantle Conditions.”

Physics of the Earth and Planetary Interiors, vol. 287, Elsevier B.V., Feb. 2019, pp.

65–75, doi:10.1016/j.pepi.2018.12.005.

Riet, Adriaan A., et al. “The Interplay of Structure and Dynamics at Grain Boundaries.”

The Journal of Chemical Physics, vol. 149, no. 19, American Institute of Physics

Inc., Nov. 2018, p. 194501, doi:10.1063/1.5052188.

Rupert, T. J., et al. “Experimental Observations of Stress-Driven Grain Boundary

Migration.” Science, vol. 326, no. 5960, American Association for the Advancement

of Science, Dec. 2009, pp. 1686–90, doi:10.1126/science.1178226.

Sastry, S. “The Relationship between Fragility, Configurational Entropy and the Potential

Energy Landscape of Glass-Forming Liquids.” Nature, vol. 409, no. 6817, Nature

Publishing Group, Jan. 2001, pp. 164–67, doi:10.1038/35051524.

Sastry, Srikanth, Pablo G. Debenedetti, Frank H. Stillinger, et al. “Potential Energy

Landscape Signatures of Slow Dynamics in Glass Forming Liquids.” Physica A:

Statistical Mechanics and Its Applications, vol. 270, no. 1, Elsevier Science

Publishers B.V., Aug. 1999, pp. 301–08, doi:10.1016/S0378-4371(99)00259-9.

---. “Signatures of Distinct Dynamical Regimes in the Energy Landscape of a Glass-

Forming Liquid.” Nature, vol. 393, no. 6685, Nature Publishing Group, June 1998,

pp. 554–57, doi:10.1038/31189.

Sayle, D. C. “Strain Reduction in Supported Materials; the Formation of Cracks and

Dislocations.” Journal of Materials Chemistry, vol. 9, no. 2, ROYAL SOC

CHEMISTRY, Feb. 1999, pp. 607–16, doi:10.1039/a807377e.

106

Saylor, David M., et al. “Distribution of Grain Boundaries in Magnesia as a Function of

Five Macroscopic Parameters.” Acta Materialia, vol. 51, no. 13, Elsevier, Aug.

2003, pp. 3663–74, doi:10.1016/S1359-6454(03)00181-2.

Sellers, Michael S., et al. “β-Sn Grain-Boundary Structure and Self-Diffusivity via

Molecular Dynamics Simulations.” Physical Review B, vol. 81, no. 13, Apr. 2010, p.

134111, doi:10.1103/PhysRevB.81.134111.

Sempolinski, D. R., and W. D. Kingery. “Ionic Conductivity and Magnesium Vacancy

Mobility in Magnesium Oxide.” Journal of the American Ceramic Society, vol. 63,

no. 11–12, Nov. 1980, pp. 664–69, doi:10.1111/j.1151-2916.1980.tb09857.x.

Shannon, Robert D. “Revised Effective Ionic Radii and Systematic Studies of Interatomic

Distances in Halides and Chalcogenides.” Acta Crystallographica, Section A, vol.

32, Mar. 1976, pp. 751–67, doi:10.1107/s0567739476001551.

Shvindlerman, L. .., and B. .. Straumal. “Regions of Existence of Special and Non-

Special Grain Boundaries.” Acta Metallurgica, vol. 33, no. 9, Sept. 1985, pp. 1735–

49, doi:10.1016/0001-6160(85)90168-3.

Sørensen, M. R., et al. “Diffusion Mechanisms in Cu Grain Boundaries.” Physical

Review B, vol. 62, no. 6, Aug. 2000, pp. 3658–73.

Speziale, Sergio, et al. “Quasi-Hydrostatic Compression of Magnesium Oxide to 52 GPa:

Implications for the Pressure-Volume-Temperature Equation of State.” Journal of

Geophysical Research: Solid Earth, vol. 106, no. B1, 2001, pp. 515–28,

doi:10.1029/2000JB900318.

Stillinger, Frank H., and Thomas A. Weber. “Packing Structures and Transitions in

Liquids and Solids.” Science, vol. 225, no. 4666, American Association for the

107

Advancement of Science, 1984, pp. 983–89, doi:10.1126/science.225.4666.983.

Stubican, Vladimir S., and John W. Osenbach. “Influence of Anisotropy and Doping on

Grain Boundary Diffusion in Oxide Systems.” Solid State Ionics, vol. 12, Elsevier

{BV}, Mar. 1984, pp. 375–81, doi:10.1016/0167-2738(84)90167-X.

Sun, Xiaowei, et al. “Properties of MgO at High Pressures: Shell-Model Molecular

Dynamics Simulation.” Physica B: Condensed Matter, vol. 370, no. 1, 2005, pp.

186–94, doi:https://doi.org/10.1016/j.physb.2005.09.011.

Sutton, A. P., and R. W. Balluffi. “Interfaces in Crystalline Materials.” Monographs on

the Physics and Chemistry of Materials, no. 51, Oxford University Press, 1995,

https://global.oup.com/academic/product/interfaces-in-crystalline-materials-

9780199211067?cc=co&lang=en&.

Suzuki, A., and Y. Mishin. “Atomic Mechanisms of Grain Boundary Diffusion: Low

versus High Temperatures.” Journal of Materials Science, vol. 40, no. 12, June

2005, pp. 3155–61, doi:10.1007/s10853-005-2678-0.

Tschopp, M. A., et al. “Binding Energetics of Substitutional and Interstitial Helium and

Di-Helium Defects with Grain Boundary Structure in α-Fe.” Journal of Applied

Physics, vol. 115, no. 3, American Institute of Physics, Jan. 2014, p. 033503,

doi:10.1063/1.4861719.

Upmanyu, M., et al. “Simultaneous Grain Boundary Migration and Grain Rotation.” Acta

Materialia, vol. 54, no. 7, Apr. 2006, pp. 1707–19,

doi:10.1016/j.actamat.2005.11.036. van Beest, B. W. H., et al. “Force Fields for Silicas and Aluminophosphates Based on Ab

Initio Calculations.” Physical Review Letters, vol. 64, no. 16, American Physical

108

Society, Apr. 1990, pp. 1955–58, doi:10.1103/PhysRevLett.64.1955.

Van Orman, J. A., and K. L. Crispin. “Diffusion in Oxides.” Reviews in Mineralogy and

Geochemistry, vol. 72, no. 1, GeoScienceWorld, Jan. 2010, pp. 757–825,

doi:10.2138/rmg.2010.72.17.

Van Orman, James A., et al. “Aluminum Diffusion and Al-Vacancy Association in

Periclase.” Physics of the Earth and Planetary Interiors, vol. 172, no. 1–2, Jan.

2009, pp. 34–42, doi:10.1016/j.pepi.2008.03.008.

Van Orman, James A, et al. “Diffusion in MgO at High Pressures: Constraints on

Deformation Mechanisms and Chemical Transport at the Core-Mantle Boundary.”

Geophysical Research Letters, vol. 30, no. 2, Jan. 2003, p. 1056,

doi:10.1029/2002GL016343.

Verma, Ashok K., and Bijaya B. Karki. “First-Principles Simulations of MgO Tilt Grain

Boundary: Structure and Vacancy Formation at High Pressure.” American

Mineralogist, vol. 95, no. 7, July 2010, pp. 1035–41, doi:10.2138/am.2010.3386.

Vincent-Aublant, E., et al. “Self-Diffusion near Symmetrical Tilt Grain Boundaries in

UO2 Matrix: A Molecular Dynamics Simulation Study.” Journal of Nuclear

Materials, vol. 392, no. 1, North-Holland, July 2009, pp. 114–20,

doi:10.1016/j.jnucmat.2009.03.059.

Webb, S., and I. Jackson. “Anelasticity and Microcreep in Polycrystalline MgO at High

Temperature: An Exploratory Study.” Physics and Chemistry of Minerals, vol. 30,

no. 3, Apr. 2003, pp. 157–66, doi:10.1007/s00269-003-0299-1.

Wicks, J. K., et al. “Very Low Sound Velocities in Iron-Rich (Mg,Fe)O: Implications for

the Core-Mantle Boundary Region.” Geophysical Research Letters, vol. 37, no. 15,

109

Aug. 2010, p. n/a-n/a, doi:10.1029/2010GL043689.

Wuensch, B. J., and T. Vasilos. “Diffusion of Transition Metal Ions in Single-Crystal

MgO.” The Journal of Chemical Physics, vol. 36, no. 11, 1962, pp. 2917–22,

doi:10.1063/1.1732402.

---. “Grain-Boundary Diffusion in MgO.” Journal of the American Ceramic Society, vol.

47, no. 2, Feb. 1964, pp. 63–68, doi:10.1111/j.1151-2916.1964.tb15656.x.

Yamamoto, R., and M. Doyama. “The Interactions between Vacancies and Impurities in

Metals by the Pseudopotential Method.” Journal of Physics F: Metal Physics, vol. 3,

1973, p. 1524.

Yang, M. H., and C. P. Flynn. “Intrinsic Diffusion Properties of an Oxide: MgO.”

Physical Review Letters, vol. 73, no. 13, American Physical Society, Sept. 1994, pp.

1809–12, doi:10.1103/PhysRevLett.73.1809.

Zha, C. S., et al. “Elasticity of MgO and a Primary Pressure Scale to 55 GPa.”

Proceedings of the National Academy of Sciences, vol. 97, no. 25, National

Academy of Sciences, Dec. 2000, pp. 13494–99, doi:10.1073/pnas.240466697.

Zhang, Liqun, et al. “Molecular Dynamics Investigation of MgO-CaO-SiO2 Liquids:

Influence of Pressure and Composition on Density and Transport Properties.”

Chemical Geology, vol. 275, no. 1–2, June 2010, pp. 50–57,

doi:10.1016/j.chemgeo.2010.04.012.

110