Reactive Force-Field Validation
of Diffusion Properties in Ceria Nanoparticles
Bachelor Thesis
Author:
Dennis Larsson
Supervisor:
Dr. Jolla Kullgren
Department of Chemistry - Ångström Laboratory
Structural Chemistry
April 7, 2016
Sit down before fact as a little child, be prepared o give up every preconcieved notion, follow humbly
wherever and to whatever abysses nature leads, or you shall learn nothing - Thomas Henry Huxley
Sammanfattning: Ceria (CeO2 ) är en övergående jonisk förening och ett material med en rad
tekniska användningsområden. I många av dessa tillämpningar spelar syrevakanser (dvs. luckor i
syregittret) en avgörande roll, t.ex. vid jondiffusion och katalys. I detta projekt utfördes studier av s.k.
"reaktiva kraftfält" och speciellt deras förmåga att beskriva egenskaper kopplade till syrevakanserna,
såsom syrediffusionen.
Kraftfältet hade parameteriserats av Broqvist et al för denna typ av studier. Som det kommer att
framgå av den här rapporten så existerade det defekter i kraftfältet som krävde att kraftfältet omparameterisedes och omvaliderades. Två dotterkraftfält undersöktes genom att avgöra deras förmåga att
beskriva vakansegenskaper och diffusion för syre i Ceria. Inget av dotterkraftfälten visade sig kapabelt
att beskriva dessa egenskaper tillräckligt och vidare omparametrisering kommer därför att krävas.
Abstract: Ceria (CeO2 ) is a primarily ionic compound and a material with a wide area of technical
applications. In many of these applications oxygen vacances (e.g. absent oxygens in the oxygen lattice)
play a decisive part, for example in ionic diffusion and catalysis. In this project studies of so called
"reactive force-fields" was performed and their capabilities to describe properties related to the oxygen
vacancies, such as oxygen diffusion.
The force-field had been parameterized by Broqvist et al for this type of studies. However, as will
be made clear by this report the force-field was imperfect and re-parameterization and re-validation of
the force-field was necessary. Two derivative trial force-fields was explored by determining their performance in describing oxygen vacancy energetics and oxygen diffusion properties. Unfortunately
neither force-field was capable of adequately describing these properties, necessitating further reparameterziation.
Contents
1 Introduction
1.1 Structure of Ceria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Diffusion Model of Ceria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
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2 Methods in Computational Chemistry
2.1 Interaction Models . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Density Functional Theory . . . . . . . . . . . . . . .
2.1.2 Classical Force-Fields . . . . . . . . . . . . . . . . . .
2.1.3 Reactive Force-Fields . . . . . . . . . . . . . . . . . .
2.2 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . .
2.3 Diffusion Calculations from Molecular Dynamics Simulations
2.4 Calculations of Barriers . . . . . . . . . . . . . . . . . . . . .
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3 Calculations in this project
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3.1 Force-Field Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Results and Discussion
4.1 Assesment of the Original ReaxFF . . . . . .
4.2 Assessment of FF1 and FF2 - Bulk properties
4.3 Assessment of FF1 and FF2 - Energetics . . .
4.4 Assessment of FF1 and FF2 - Diffusion . . .
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5 Conclusions
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Acknowledgement
18
References
18
1
Introduction
Cerium (Ce) is the most common of the lanthanides in the Earth’s crust, being the 25th most abundant element there, comparable with the concentration of Copper [1]. A unique feature of Cerium
among the lanthanides is the capacity to exist in two stable oxidation states, 3+ and 4+ [2]. The
capacity for Cerium to remain stable in two oxidation states allows Cerium to easily be reduced in reductive environments and re-oxidised in oxidizing environments. This property of Cerium is exploited
in applications of Ceria, CeO2 , and allow Ceria to exchange oxygen with its surrounding through formation and anhilation of oxygen vacancies. The vacancy formation process is given by the reaction:
2Celatt4+ + Olatt2 – ! 2Celatt3+ + 12 O2 (g). Thus upon releasing molecular oxygen, and in the process
generating an oxygen vacancy, two Ce4+ are reduced to Ce3+ . This, in combination with the fluorite
structure of Ceria, which is a structure that contains large and empty octahedral sites [3], allows for
fast oxygen diffusion and makes Ceria an excellent material for application in solid oxide fuel cells
(SOFCs) and in catalytic converters in automobiles [4–7]. In the former Ceria acts as a membrane
separating a reductive environment from an oxidizing environment. On the surface exposed to the
oxidizing environment oxygen molecules are absorbed into Ceria while oxygen is released into the reductive environment. In the case of SOFCs the oxidation typically involves the oxidation of H2 into
H2 O as shown schematically in figure 1.
Hydrogen
Oxygen
Vacancies
CeO2
Water
Oxygen ions
Anode
Cathode
Figure 1: Schematic image of a SOFC. The first magnification shows agglomerated Ceria octahedra
from experiment. The second magnification shows a structural model of a twin Ceria nanoparticles
intended for simulating grain boundary environment in the agglomeration. Red spheres are oxygen ions,
the black ones are Cerium and the white ones are hydrogen. TEM image adopted from reference [8].
In SOFCs, we can image the vacancies as the moving (diffusing) "object" instead of the oxygen
ions in a process starting when oxygen is released from Ceria on the anodic side, then through the
electrolyte to the cathodic side where new oxygen is absorbed. The Ceria electrolyte in SOFCs consists
of agglomerated Ceria nanoparticles. This means that during the process of moving from anode to
cathode the vacancies will have to travel trough different structural environments such as through bulk,
along surfaces and over the interfaces between different nanoparticles (also known as grain boundaries).
This creates challenges for studying the diffusion mechanism and which computational method to use.
While density functional theory (DFT, section 2.1.1) might be capable of giving a description of the
bulk and surface mechanisms, the complexity of grain boundaries currently makes it impossible because
the size and complexity of the problem. Force-fields (2.1.2)are in principle capable of describing the
1
movement of vacancies but not any related reactive properties, such as the interconversion between
Ce4+ and Ce3+ which are treated as two distinct types of partices.
In order to overcome these shortcomings of DFT and force-fields this study will utilise what is
known as a reactive force-field (ReaxFF, section 2.1.3) [9–11] to try to capture the essentials of the
diffusion mechanisms in Ceria. The particular force-field used was parameterized by Broqvist et al [12].
1.1
Structure of Ceria
Ceria features a cubic fluorite structure (space group Fm3m). Describing this structure is most easily
done with a Bravais lattice, which gives the positions of the atoms as fractions of the cell axes in the
(x,y,z) Cartesian directions as indicated in figure 2. When referring to the length of a cell axis it is
customary to call them cell parameters, also denoted (a,b,c). In the cubic case, all axes are equal and
are denoted a. Other cell parameters often considered are the angles between the three axes, (↵, , ).
However, these will be completely omitted in this work since all structures considered have orthogonal
axes.
0.5.5
.50.5
000
Z
.5.50
y
x
Figure 2: Bravais lattice points for a cubic F-cenetered lattice.
In the fluorite structure, the cerium ions adopt a F-centered lattice, meaning that they are positioned at (0,0,0), ( 12 , 12 ,0), (0, 12 , 12 ), ( 12 ,0, 12 ). This is to be interpreted as (0a,0a,0a), ( 12 a, 12 a,0a),
(0a, 12 a, 12 a), ( 12 a,0a, 12 a). Here it is worthwhile to note that when placing atoms in the unit cell, positions 0 and 1 are often written togheter, so position (0,0,0) indicates that atoms are placed at (0,0,0),
(1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1) and (1,1,1). This is done since these positions are equivalent by translational symmetry and contribute equally to the total formula unit of the cell. The eight
positions given by (0,0,0) all contribute 1/8 atom giving a total of 1 atom to the cell since they are
corner positions and thus shared between eight unit cells if the system is expanded.
When considering where the oxygen ions are positioned in the unit cell it helps to think about
empty sites within the cell as geometrical shapes formed by taking the cerium ions as corners. In a
F-centered lattice this will result in two different types of sites called tetrahedral sites (which there are
eight of) and one octahedral site, see figure 3.
2
(a) F-centered
Ce lattice
(b) Tetrahedral site
(c) Octahedral site
Figure 3: Figure of (a) the F-centered Cerium lattice, (b) a tetrahedral site and (c) the octahedral
site.
Since an atom placed in either of these sites contribute one whole atom to the crystallographic unit
cell and electroneutrality in the cell needs to be achieved, one can simply sum the charge given by the
cerium lattice to realize that each tetrahedral site needs to contain one oxygen ion, giving the oxygen
positions as ( 14 , 14 , 14 ), ( 34 , 14 , 14 ), ( 14 , 34 , 14 ),( 34 , 34 , 14 ), ( 14 , 14 , 34 ), ( 34 , 14 , 34 ), ( 14 , 34 , 34 ) and ( 34 , 34 , 34 ) generating the
Ceria fluorite structure in figure 4.
Figure 4: Ceria fluorite structure with black Cerium ions and red Oxygen ions. Figure adopted from
reference [12].
Nanoparticles are essentially large crystals built up by adding several of these unit cells into one
large cell. However, since Ceria nanoparticles are not cubic, but in fact octahedral as shown in figure 1,
a way to cut the unit cells needs to be introduced. This is done by so called lattice planes given by
Miller indices, defined as (hkl) where h, k and l corresponds to the inverted x, y and z points where
the planes cuts the axes. In other words,
h=
1
1
1
, k= , l=
x
y
z
(1)
so the plane (222) would be the plane that cuts the axes at half cell length. Figure 5 shows low-index
lattice planes (100), (110) and (111).
3
Figure 5: Figure of lattice planes (a) (100), (b) (110) and (c) (111).
Bulk (unit cell) and surfaces (lattice planes) features different chemical environments as a result of
the different bond order, or coordination, present. The coordination number of an atom or ion can be
viewed as the amount of atoms or ions that it forms a bond with. From figure 4 one can deduce that
in bulk each cerium ion has eight oxygen ions as their closest "neighbor", while each oxygen ion has
four cerium ions as neighbors, giving the total unit cell coordination as 8:4. In the surface regions the
total coordination will be reduced as a result of "removing" atoms. At the nanoparticle boundaries, or
grain boundaries as they are commonly known, the coordination is much more complicated and won’t
be covered any further in this report.
Theoretical estimates of surface energies of Ceria suggest the ordering of stable surfaces as (111)
being the most stable surface followed by (110) with (100) being the least stable. This is consistent
with experimental findings typically showing octahedral nanoparticles which exposes the (111) surface.
How to construct a octahedral ceria nanoparticle is show in figure 6.
(a) Ceria(111)
(b) NP surface
(c) NP with bulk
content
Figure 6: Figure (a) Ceria(111), (b) Ceria nanoparticle surface and (c) a nanoparticle with bulk
content. Colouring as in previous figures. (c) is adopted from reference [12].
However, perfect octahedral particles of Ceria are always non-stoichiometric. To create stoichiometric nanoparticles the edges need to be truncated, exposing some (100) surface, see figure 7.
4
Figure 7: Figure of a truncated octahedral Ceria nanoparticle with exposed (100). Colouring as in
previous figures. Figure adopted from reference [12].
As mentioned earlier, the application of Ceria in SOFCs is based on the materials defective nature
with a large amount of oxygen vacancies present. When vacancies are introduced, oxygen lattice
positions are vacated as shown in figure 8. As a result of the introduction of a vacancy, it becomes
possible for one of the surrounding oxygen ions to occupy it through diffusion.
Figure 8: Figure of a Ceria unit cell with one oxygen vacancy. Gray spheres here represent Ce3+ , black
Ce4+ and the red are Oxygen ions with a vacancy in white with a dashed border.
1.2
Diffusion Model of Ceria
While atomic movements in the solid state may seem counter-intuitive it actually occurs in all inorganic
materials (metals, ceramics) at all temperatures above 0 K. There are in general two types of diffusive
process, interstitial and substitutional. Interstitial diffusion requires less energy for the process to begin
and normally occurs in so called solid solutions where an impurity occupies a normally unoccupied site
in the crystal.
Substitutional diffusion on the other hand requires more energy than interstitial diffusion and is
completely dependent on the presence of vacancies. In Ceria, the absence of an oxygen frees one of
the rtetrahedral sites such that one of the neighboring oxygens can take its place.
For theoretical studies of nanoparticles with ReaxFF we could take (at least) two different routes.
One is to study the particle piece-wise by using surface cuts (called slabs) and bulk supercells to
simulate the chemical properties of different parts of the particles in individual simulations. This
however leads to errors, both small and large, in the calculations due to the lack of interconnection
5
between the different parts of the particle as each is treated separately. The other route is to treat
the nanoparticle as a whole. While this could appear to be the preferred strategy it comes with a
computational cost. Furthermore, as will be demonstrated in this report: when using an approximate
method such as ReaxFF, small errors in the description of bulk and surfaces can be accumulated in
nanoparticles since they consist of said parts.
Here both strategies are adopted and we are using both single nanoparticles and pairs of nanoparticles (figure 9) to study the diffusive properties of oxygen in and between Ceria nanoparticles in an
effort to simulate, realisticaly, key processes in the ion diffusion in the SOFC. In the study we find
that although the original force-field described in ref [12] works really well for static calculations such
as structural optimisations and vacancy formation energetics, the presence of false minima in the potential energy landscape leads to erroneous results for dynamic properties. As a consequence, further
study with this original force-field was set aside in favour of evaluating two new force-fields.
Figure 9: Image of a pair of truncated Ceria nanoparticles.
2
Methods in Computational Chemistry
There are a great number of available methods in modern computational chemistry. While ab initio
quantum mechanical (QM) methods produces the most pure results, the severe increase in computing
time with increasing system sizes means that most computers are incapable of treating systems with
up to just a few hundred atoms. This is of course problematic when applying computational chemistry
to the solid state since solid particles are formed from several thousands of atoms at the very least.
Where quantum mechanical methods suffer from too high demand in computational power, atomistic methods (or force-field methods) can be introduced as a way of simplifying the system. In forcefields (FFs) electrons are neglected in favour of retaining atoms as the smallest unit in consideration.
Furthermore, force-fields consider atoms as classical particles, more specifically as point charges with
mass, subjected to Newtonian mechanics. This is of course a severe approximation as reactive properties, such as the interconversion between Ce3+ and Ce4+ , are completely lost. However, Newtonian
mechanics are far less computationally demanding than quantum mechanics making force-fields applicable to much large systems consisting of several thousands of atoms. When treating some dynamic
properties such as diffusion which are less dependent upon actual electronic interactions and more on
electrostatic interactions between the atoms in the crystal this approximation becomes acceptable.
Should the system of interest be increased further, even atomistic methods becomes too computationally demanding, and mesoscopic (coarse-grained) models, or even continuum models, have to be
introduced. In continuum models, the material is considered to be constituted of a continuous mass
that completely fills the space it occupies. Figure 10 illustrates the relationship between different
computational chemistry methods and the size of the systems they can deal with.
6
Figure 10: Simplified scheme of computational chemistry methods along the time and size scales that
they can handle.
2.1
Interaction Models
This section will expand upon the computational methods relevant for the project. These are DFT,
classical force-fields and of course, ReaxFF, which is an advanced classical force-field. While the
actual calculations and results in this project are solely based on ReaxFF, a good or at least a minimal
understanding of the pro et contera of classical force-fields and DFT are necessary to explain the unique
and useful nature of ReaxFF.
2.1.1
Density Functional Theory
At the dawn of quantum-chemistry ab initio quantum mechanical methods was un-applicable to solid
state models. The first quantum-chemical methods features such severe scaling with system size that
they were confined to studying systems of only a handful of atoms. Several attempts were made to
introduce ab initio methods to the solid state but the breakthrough did not arrive until 1965 when
Kohn and Sham introduced modern DFT [13], which remains the most widely used computational
method for the solid state in academia. In DFT, the energy of a structure is treated as a functional of
the overall electron density of the structure instead of as a function of explicit electrons.
The electron density, ⇢(r) is defined with Kohn-Sham orbitals, m (r), as
X
⇢(r) =
| m (r)|2
(2)
m
the Kohn-Sham orbitals are obtained through the so called Kohn-Sham equations which are Schrödinger
like equations where the Hamiltonian is treated as a sum of three components.

Z
⇢(2)
ĥ(1) +
d⌧2 + Vxc (1) m (1) = ✏m m (1)
(3)
r12
Equation 3 is solved iteratively. For each iteration, starting from a qualified guess for the initial
orbitals (called basis set), the obtained orbitals are used in solving the next iteration. This is continued
until the orbitals of successive iterations no longer vary. Since the Kohn-Sham equations are solved
until the orbitals are in effect constant they are said to be solved self-consistently.
7
The Kohn-Sham Hamiltonian contains the exchange-correlation potential Vxc which is the main
source of concern for DFT methods. Since there are many ways to treat this potential, there are a
great number of different DFT functionals to choose from when applying DFT to a structure.
2.1.2
Classical Force-Fields
Classical force-fields are atomistic models, meaning that atoms are considered the smallest unit of
the system under observation instead of the electron [14] which is normally the object of interest in
chemistry. As mentioned in section 2 atoms are considered as classical particles subject to Newtonian
mechanics within force-field methods. This allows for fast calculations and can be applied to very large
systems of several thousand atoms. However, the loss of the electrons means that a detailed study
of reaction mechanisms taking place during the dynamic simulation, the breaking and re-forming
of bonds, becomes impossible. Also, the important redox properties of a material such as Ceria is
lost since Ce4+ and Ce3+ are typically treated as two completely different atoms without any redox
capabilites. A proper study of these phenomena would require the use of a QM approach, which is far
too computationally demanding to deal with systems in large scale dynamic simulations.
The energy of a classical force-field is at its most basic level a sum of 5 terms, three which are
considered to be bonding terms and the remaining two are considered to be non-bonding
E = Estretch + E✓ + E + EvdW + Eelec
(4)
The three first terms in equation 4 are the three bonding terms that describe how the energy of a bond
is affected by stretching, bending and torsion. The last two terms are the non-bonding terms which
are terms for the van der Waals attraction and electrostatic interactions between the atoms.
2.1.3
Reactive Force-Fields
ReaxFF attempts to overcome the short-coming of non-reactive classical force-fields in describing
electrons and chemical reaction phenomena through a parameterization routine where an implicit
description of the electrons in the system is added to the energy terms in equation 4 by a quantum
mechanical approach, in general DFT. A training set consisting of QM-derived data is used to fit
the force-field model. Some examples of parameters that can be used in the training set are the
atomic charges, bond dissociation energies, bond lengths, angle strains, heats of formation, vibrational
frequencies and transition state energies. The force-field is then validated for the chosen parameters
and if the force-field reproduces QM and/or experimental results to a satisfying degree it is deemed
sufficient for usage in dynamical simulations.
The end result is in principle a classical force-field model capable of dealing with both dynamical
properties of the system as well as containing a crude description of the reactions taking place during
the simulation process. One important drawback to take into consideration with ReaxFF is that the
method lacks adaptability and is only capable of generating results for the parameters included in the
training set. A ReaxFF trained for diffusion simulations in ZrO2 would not be able to describe the
diffusion processes in CeO2 .
2.2
Molecular Dynamics
In molecular dynamics simulations, Newton’s laws are applied to each atom in the system. The atoms
in the system are allowed to interact through a potential energy surface which creates a realistic model
of atomic motions, which in this work was generated using ReaxFF. Examples of application of these
types of simulations in the literature include crystal growth, liquid evaporation and, as shown in this
report, the diffusion of atoms in the solid state.
By describing the motion of each atom with Newton’s first law of mechanics
@ 2 ri (t)
=
@t2
1
r (ri (t))
mi
8
(5)
where ri (t) is the position of atom i at time t, mi the mass of atom i and r (ri (t)) is the instantaneous
force on atom i. By simultaneously integrating equation 5 for each atom in the system the trajectory
of each atom i is obtained. The integration method used here was the Velocity Verlet algorithm [15],
which yields the position, velocity and acceleration of each atom i upon integration, allowing the state
of the system in the future to be predicted.
From the trajectory, the mean square displacement of the atoms can be calculated by
hr2 i =
1
Natom
NX
atom
i
h|ri (t)
ri (t0 )|2 i
(6)
where ri (t0 ) is the position of atom i at the start of the simulation. The mean square displacement
of the atoms can then be used to calculate the diffusion of the atoms as explained in the following
section.
2.3
Diffusion Calculations from Molecular Dynamics Simulations
Diffusion in Ceria is the movement of oxygen ions in understoichiometric CeO2-x (CeO2-x is the common
notation for Ceria with missing oxygen ions, i.e. oxygen vacancies) from one oxygen position to another
in the crystal lattice. By calculating the mean square displacement of the moving atoms, the diffusion
constant, D, is obtained by
hr2 i = 6Dt
(7)
where t stands for the time of the displacement.
The diffusion is related to the energy required to complete the motion, Ea , by an Arrhenius-type
expression
Ea
D = D0 e RT
(8)
By taking the logarithm of equation 8 one obtains
ln(D) = ln(D0 )
Ea
RT
(9)
which is a useful expression for obtaining both the diffusion pre-factor D0 and the activation energy
by plotting the logarithm of the diffusion against the reciprocal of the temperature at which the
displacement took place.
We propose that the diffusion of oxygen in Ceria is activation controlled, meaning that so long as
there is enough energy to start the process, none or very small kinetic barriers will impede the diffusion
rate. To validate this theory, the activation energy in equations 8 and 9 will be compared to transition
state barriers obtained by nudged elastic band (NEB) calculations, which will be expanded upon in
the following section.
2.4
Calculations of Barriers
Transition State Theory (TST) makes prediction as to the forming and breaking of an activated
complex during chemical reactions. In general, a reaction A + B
AB is considered to first go to
a transient structure, T S ‡ , which will then either break apart into the reactants or form the chemical
product. So within TST the before-mentioned reaction is proposed to follow A + B
T S‡
AB. An important feature of TST is that in addition to just considering the thermodynamics of a
reaction, the kinetics is explored as well, which in TST follows an Arrhenius-type expression.
In this work, we propose that the movement of an oxygen atom from one lattice position to a
vacancy follows this theory. At the midpoint between the initial oxygen position and the vacancy, a
peak energy will be reached and the ion will either fall back to its original position or continue on to
the vacancy.
9
The activation energy for these movements, "the barriers", will be calculated by nudged elastic
band (NEB) calculations of the displacement of an oxygen ion. The NEB algorithm follows a simple
routine where the atom or ion under observation is moved from one position to another in a series
of steps. These steps are then connected by addition of an artificial force between the images like
a rubber band. For symmetrical structures such as fluorite, the TST barrier for bulk is expected to
appear at the midpoint between the the start and endpoints. For that reason, all bulk barriers was
calculated using odd-integer steps. A simplified three step NEB is shown in figure 11.
Figure 11: Simplified picture of a three step NEB procedure with (a) the initial structure with an
oxygen vacancy in the upper right part if the figure, (b) an intermediary transition state and (c) the
final structure with the vacancy to the upper left.
Should our hypothesis be correct, the activation energy as obtained from diffusion values and
NEB calculation should be close to each other. Also, since both theories follows an Arrhenius type
expression, the pre-factors should be equal in magnitude. Due to time constraints the latter part was
unfortunately not explored and focus was put on calculating the barriers.
3
Calculations in this project
This section expands briefly on the calculations performed during the course of the project by describing
the structures and the software that was utilised.
3.1
Force-Field Calculations
All ReaxFF calculations was performed within the Atomic Simulation Environment (ASE) platform
[16] with the FIRE optimizer [17] calling the LAMMPS software [18]. Periodic boundary conditions
were applied in three dimensions. Oxygen vacancies were introduced in nanoparticles and surface
slabs by moving one or more of the oxygen atoms into the vacuum more than 10 Å from the structures
surface. This was due to a technicality in the ReaxFF program, namely the use of a tapering function
for the electrostatic interactions in ReaxFF, meaning that for distances above 10 Å all electrostatic
interactions are set to zero. In bulk supercells oxygen vacancies were created by removing one or more
of the oxygen atoms.
Bulk properties were calculated by performing energy-volume scans for both the fluorite and rutile
polymorphs of Ceria, displayed in figure 12. Rutile was included in order to check that it was not artificially overstabilized with the force-field, ensuring that fluorite remained the most stable polymorph.
The Travis program [19] was used to calculate Ce-O bond lengths.
10
Figure 12: Bulk crystal structures of (a) fluorite CeO2 and (b) rutile CeO2 . Figure adopted from
reference [12].
3.2
Molecular Dynamics Simulations
MD simulations were run on bulk supercells made out of 8392 atoms (CeO1.8778 ) within LAMMPS at
different temperatures using an NPT-ensemble. The Velocity Verlet algorithm was used to initiate the
thermostating of the supercell. The simulations were run for 45 ps at 1 fs timesteps with 10 fs sampling
rate. When calculating the mean square displacement of the atoms the first 20 ps were neglected. This
is due to the large oscillation of the atoms during this time period, which can be considered to be the
time required to equilibrate the system at the specified temperature. The Travis program was then
used to calculate the mean square displacement and the diffusion of the atoms.
4
Results and Discussion
The following section will present the computational results from the project. The section is divided
into four subsections. Section 4.1 will describe the strengths of the original potential by Broqvist et al.
as well as illuminate the fundamental complications with the potential which led to it being abandoned.
Sections 4.2, 4.3 and 4.4 will present the potential comparisons that dominated this project and will
showcase how the force-fields compare when describing bulk properties, energetics and the diffusion in
Ceria respectively.
4.1
Assesment of the Original ReaxFF
As described thoroughly by Broqvist et al., the original force-field worked remarkably well in describing
static properties in Ceria and the hope was that this would translate into a well defined description
of the diffusive properties of oxygen in Ceria. However, these properties had not been evaluated in
ref [12] so the natural question to answer for this potential was if it could describe these properties.
For this purpose, the energy for creating an oxygen vacancy in Ceria in the bulk part of the slab and
at the sub-surface and surface positions was calculated. The vacancy energy profile for first creating
an oxygen vacancy in bulk Ceria, then performing a geometry optimisation of the structure and then
moving the vacancy one oxygen layer closer to the surface while initiating a new geometry optimisation
can be seen in figure 13. While the profiles for slabs of Ceria(111) and Ceria(110) look promising and
correspond well to DFT data, reference [12], the same profile for a truncated nanoparticle in which the
vacancy was moved to a (111) surface showed a disturbing pattern where the vacancy was stabilized
upon moving from the deepest bulk positions and even more worrying the subsurface and surface
positions showed no difference when compared to the outermost bulk positions.
11
Ceria(111)
Ceria(110)
Ceria NP
0
Evac,rel [eV]
-1
-2
-3
-4
1
Bulk
2
3
Oxygen position nr i
4
5
Surface
Figure 13: Relative vacancy energies for moving an oxygen from an inner bulk-like position, i=5, to
the surface, i=1, calculated using slab and NP models described by the Broqvist et al. ReaxFF. The
models used were Ceria (111), Ceria (110) and a Ceria nanoparticle. The energy of the innermost
bulk-like positions were arbitrarily set as zero in each case and the other values were set in comparison
to these values.
This was clearly a problem since it is reasonable to assume that this strange profile would have
consequences for the oxygen diffusion in Ceria nanoparticles which we hoped to be able to study. While
no indication as to the fundamental issue of the potential can be obtained from figure 13, it posed the
follow-up question "What happens with the barrier for moving an oxygen atom from one positions
to the other?". This question has been answered in the literature by Nolan et al. [20], where a DFT
barrier was calculated to be 0.5 eV using a NEB routine. However, the potential generated a barrier
which not only did not reach much higher than 0.1 eV, it also features two extra minima along the
route, figure 14. Since these types of local minima are not present in the DFT profile, reference [20],
they were clearly anomalies in the potential which required further investigation.
By studying how the structure was distorted by the introduction of a oxygen vacancy short Ce-O
bond distances was observed after the geometry optimisation. Suspecting that a large amount of such
short bond distances could explain the NP profile in figure 13 radial distribution calculations were
performed on the nanoparticle for each of the steps in the profile. These were calculated by setting
each Cerium ion as the centre and then calculating the distances to the nearest neighboring ions in a
sphere around the ion and are shown in figure 14. The results clearly indicated that the drop in energy
and the false minima were results from a large number of short bonds.
12
E [eV]
0.1
E [eV]
0.1
0.0
0.0
0.0
1.0
2.0
0.0
Path [Å]
1.0
2.0
Path [Å]
(a) 5-step NEB barrier
(b) 9-step NEB barrier
Figure 14: Two TST barriers for the oxygen movement from one lattice position to another that clearly
visualizes the false minima present in the original force-field.
What this essentially meant was the original force-field by Broqvist et al. appears to containe
a number of false minima which resulted in short Ce-O bonds. Furthermore, the force-field proved
incapable of optimizing the structure of twin particles of Ce140 O279 -Ce140 O280 . Due to this inadequate description of barriers and the existence of false minima, a re-parametrization of the original
potential was done and two new force-fields was obtained and validated against literature values from
both experiment and DFT. The following part of this section deals with comparing the results from
all three force-fields, denoted original, force-field 1 (FF1) and force-field 2 (FF2). These force-field
parameterizations were performed by Broqvist.
40
600
g(r)
0
0
250
200
20
0
20
200
25
250
Distance
r [pm]
Figure 15: Radial distribution of Ce-O distances in geometry optimized Ce140 O279 .
13
(a) Original Potential
(b) FF1
Fluorite
Rutile
2.5
Energy (eV / CeO2)
(c) FF2
1.5
0.5
40
50
40
50
40
50
3
Volume (Å / CeO2)
Figure 16: Energy-volume scans with (a) the original force-field (b) FF1 and (c) FF2 comparing
fluorite Ceria with rutile Ceria.
4.2
Assessment of FF1 and FF2 - Bulk properties
To validate the performance of the two new force-fields we needed answers to the questions (1) "is
the structure of Ceria accurately described?", (2) "how good does the vacancy energetics compare to
DFT?" and (3) "can we obtain any insight into the diffusive properties in Ceria?". This section will
explore the first question while the following two section will deal with the second and third questions.
To ensure that the force-fields described stable structures, the bulk modulus B0 was obtained from
energy-volume scans. B0 is a thermodynamic materials property roughly corresponding to the rigidity
of the structure. The energy-volume scans were also used to obtain the cell size, here described as the
cubic cell parameter a. The energy-volume scans was performed on both fluorite and rutile Ceria as
explained in section 3.1.
Figure 16 shows the energy-volume scans for the original, FF1 and FF2 potentials. We can see
that for each potential fluorite is the most stable polymorph. In FF2, the rutile polymorph features
an odd discrepancy, figure 16c, that was not explored further since it would not affect the stability of
the fluorite polymorph. Table 1 features obtained values for the bulk moduli and cell parameters of
the three potentials and compares them to both experimental and DFT+U values.
14
Table 1: Table of bulk parameters obtained from bulk-volume scans.
a[Å]
B0 [GPa]
Reference
Original
5.49
199.45
This work
FF1
5.50
295.73
This work
FF2
5.50
272.74
This work
Expt.
5.411(1) 220(9)
Ref [21]
5.49
Ref [12]
DFT+U
(U = 5 eV)
179
In table 1 we can see that while the cell parameter changed only very little with the new force-fields,
they were far more rigid. While these results will not have any real effect on the diffusive properties of
oxygen in Ceria, they serve as a verification of the fact that fluorite Ceria is the most stable structure
with these force-fields and as such will not morph into another structure during the MD simulations.
4.3
Assessment of FF1 and FF2 - Energetics
Here the energy required to create an oxygen vacancy from both bulk and the low-index surfaces (110)
and (111) was compared for the three force fields and the energy required to move an oxygen atom
from a lattice position to the vacancy was also compared. This was done on a 96 atoms bulk supercell
and on 1944 atoms surface slabs that were 36 atom layers deep.
The (100) surface was omitted from this study despite its presence in truncated octahedral Ceria
nanoparticles due it being a polar surface. Treating polar surface slabs with ReaxFF requires reconstruction into hypothetical non-polar distortions of the surface and as such add little to the validation
of the force-field.
Surface energies, Esurf was calculated by the standard method, reference [12], namely
Esurf =
E[CeO2 ] N Ebulk
2A
(10)
where E[CeO2 ] is the energy of the stoichiometric surface slab supercell, N the number of atoms in
the surface slab supercell, Ebulk the energy per formula unit of bulk, calculated separately in a bulk
supercell, and A the exposed surface of the slab supercell taken two times since two surfaces was
exposed.
The vacancy energies, Evac was calculated as
Evac,surf = E[CeO2
x
slab + O]
E[CeO2 ]
1
De (02 )
2
(11)
where E[CeO2 x slab + O] is the total energy from the structure optimisation of a Ceria surface where
one oxygen atom had been moved at least 10 Å from the surface and De the oxygen dissociation energy.
Values for the surface energies and vacancy formation energies are displayed in table 2.
15
Table 2: Calculated surface energies and vacancy formation energies with the different force-fields and
reference DFT.
Original
FF1
FF2
DFT+U
(U = 5eV)
CeO2 (111)
Esurf [Jm
2
]
Evac [eV]
0.83
0.78
0.73
0.68
2.32
1.28
1.22
2.60
1.05
1.01
0.94
1.01
1.98
1.11
1.26
1.99
CeO2 (110)
Esurf [Jm
Evac [eV]
2
]
CeO2 (Bulk)
Evac [eV]
3.04
2.29
2.47
3.39
While the results presented in table 2 confirm that with both FF1 and FF2 the (111) surface is
the most stable, the vacancy formation energies are now much too low for both the (111) surface and
the bulk. For diffusive studies the bulk values are acceptable, since they remain high enough to negate
the possibility of spontaneously creating vacancies from the bulk. The (111) values, however, indicate
that the nanoparticle surfaces are likely to be more reduced than expected in nature. This could easily
lead to an overestimation of the amount of oxygen adsorbed onto the surface and diffused through the
electrolyte in a SOFC.
Also, the transition state barriers were calculated and the barrier calculated with FF1 is presented
in figure 17. The FF1 and FF2 barriers lacked the false minima present in the original force-field. The
main differences between FF1 and FF2 is that FF1 comes close to the literature value of 0.5 eV with
a barrier of 0.456 eV while the FF2 barrier is a bit to low at only 0.320 eV. These values are tabulated
in table 3 along with the corresponding diffusion barriers.
E [eV]
0.42
0.0
0.0
1.0
2.0
Path [Å]
Figure 17: TST barriers for FF1.
16
3.0
4.4
Assessment of FF1 and FF2 - Diffusion
The diffusion of the oxygen atoms in Ceria was compared for the three different force-fields by running
MD simulations at 1000, 1250, 1500, 2000, 2500 and 3000 K. At each temperature, the resulting
diffusion constant was plotted against the reciprocal of temperature to yield the Arrhenius-type plots
in figure 18.
22
DD [cm
[cm /s]
/s]
0.00025
D = D0 e
0.0002
-
Original potential
RT
Force-field 1
Force-field 2
0.00015
0.0001
5e-05
0
(a)
-20
ln (D)
ln(D)
ln(D)
-21
-22
E 1
ln(D)
(b)= ln(D0 ) - a
R T
-23
-24
0.0003
0.0004
0.0005
0.0007
0.0006
0.0008
0.0009
0.001
-1-1
1/T [K
1/T
[K ] ]
(b)
Figure 18: Plot of (a) equation 8 and (b) equation 9 for bulk Ceria.
From the Arrhenius plots in figure 18 the activation energy, Ea , for diffusion and the diffusion prefactors, D0 , were obtained. The activation energy was then compared to the transition state barriers
from NEB calculations in table 3.
The large discrepancy between the diffusion and NEB results for the original force-field is likely a
result of the short-distance issues that is present in the potential. The results for FF1 and FF2 on the
other hand seem indicative of our initial hypothesis that the diffusion of oxygen in Ceria is activation
determined. The high diffusion pre-factor on the other hand means that the diffusion from these forcefields are too high and as such unusable for proper theoretical studies, a re-parameterization is thus
still required to perfect a force-field for diffusive studies in Ceria.
17
Table 3: Diffusion and energy barriers from MD simulation and force-field calculations.
NEB
Ea [eV]
Ea [eV]
D0 [cm2 /s]
Reference
Original
0.118
0.246
6.7824e-05
This work
FF1
0.456
0.497
1.4973e-04
This work
FF2
0.320
0.386
1.3362e-04
This work
-
0.57
5.7e-05
Ref [22]
Buckingham
DFT+U
(U = 5 eV)
5
MD Simulation
0.53
-
-
Ref [20]
Conclusions
The results presented in this report can only come to the conclusion that none of the force-field utilised
in this report accurately described the diffusive properties of Ceria nanoparticles.
While the original force-field comes closest in terms of energetics and the diffusion pre-factor, the
local minima present in the force-field means that the diffusion mechanism is likely to be in error.
Since the mechanism is a major topic of study with these force-fields, this force-field is still in need of
re-parameterization.
Force-field 1 comes close in terms of barriers and possibly describes the actual diffusion mechanism
accurately, by assuming that the diffusion is activation controlled. However, the vacancy energetics
follow a disturbing pattern as the important (111) surface is too unstable and easily reduced, and as
such the force-field would be useless for larger dynamic simulations where a O2 -rich atmosphere is
allowed to diffuse into an agglomeration of Ceria nanoparticles. Also, the diffusion rate is too high as
indicated by the diffusion pre-factor.
Force-field 2 displays no advantages at all and as such is seemingly uninteresting.
Acknowledgement
Jag vill börja med att tacka min handledare Jolla Kullgren och Peter Broqvist för den tid ni har ägnat
åt mig och det här projektet. Det har varit ett sant nöje att genomföra det och eran hjälp och era
idéer har varit ovärderliga för att kunna genomföra det.
Jag vill även tacka Kersti Hermansson och Tomas Edvinsson som öppnade upp mina ögon för detta
intressanta och smått magiska vetenskapsområdet som är beräkningskemi.
Tack till er alla i teoroo-gruppen som bidragit med intressanta idéer, tips och annan värdefull input.
Ett stort tack ska även riktas till Terese Becker och Oli Gråhed, mina grundskole- och gymnasielärare i kemi. Hade det inte varit för er och era skickliga och underhållande lektioner hade
jag aldrig suttit där jag sitter idag.
Tack till min familj, min släkt och min sambo. Jag älskar er alla.
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