Discovery

18
he reaction of hydrogen with
uranium and plutonium is
important to both the nuclear power
and nuclear weapons communities
1
.
Given this fact, it is surprising how
poorly understood this reaction actually
remains. Although experimental work
can provide us with valuable insight
into the reaction mechanism, it can
never provide the full picture. To answer
questions such as ‘which species of
hydrogen diffuses to the metal surface’
requires the use of molecular modelling.
Here, we will illustrate some of these
molecular modelling techniques and
their application to uranium hydriding.
Even ‘noble’ metals such as platinum
develop some form of oxide coating
upon exposure to an oxygen-containing
atmosphere. The phase and composition
of the surface oxide layer can greatly
affect the reaction of the underlying
metal with gases, e.g. the oxide film on
the surface of aluminium acts as a barrier
to oxygen transport through to the
metal surface and so rapidly passivates
Modelling at AWE
T
the material towards further oxidation.
Thus, metal oxides are not only important
as the external surfaces of most metals,
but also as materials in their own right.
In order to quantify the hydrogen-uranium
reactions, it is necessary to understand
the mechanisms involved in gas transport
through the surface oxide layers.
Specifically, one requires knowledge
of the species of the gas involved in
diffusion through the oxide, the diffusion
rate and its temperature dependence,
the solubility of the gas species in the
oxide and the mechanism by which it
diffuses, e.g. via interstitial sites, vacancies
or grain-boundaries within the oxide.
Our modelling studies here at AWE, at
Reading University and at Imperial
College London are aimed at answering
these questions using some of the
methods explained below.
Ab initio Calculations
The most desirable approach to
modelling a system involves using a set
of physical constants, fundamental
equations and basic assumptions and
predicting physical and chemical
properties of the system in an ab initio
manner without regard to experimental
data. In this scenario, the rôle of
experimental data is to provide a
means by which modelling results may
be validated, and not to predicate the
modelling. This can be achieved if one
uses methods based on quantum
mechanics, which enable the calculation
of structural, electronic and thermodynamic
properties of materials and quantum
molecular dynamics, which enable
transport properties and other time and
temperature-dependent properties to
be calculated. Each of these techniques
has proven to be a very powerful predictive
tool when applied to small systems
containing only a relatively low number of
electrons. However the associated
calculations rapidly become impossible
when the number of electrons increases
beyond approximately 500-1000. In fact,
quantum molecular dynamics calculations
can even become intractable problems
when this number is much smaller.
Molecular
Discovery • The Science & Technology Journal of AWE
19
In addition to the problems encountered
with systems containing a relatively
large number of electrons, actinides
contain f electrons that are problematic
in their own right for a number of reasons:
f electrons are quite diffuse in
nature, i.e. their associated
electron density is spread relatively
evenly over a large radius from the
nucleus, making overlap and
consequent interaction with other
atoms and ions weak;
partially filled f shells lead to
low-lying unoccupied states and
these cause convergence problems;
the f electrons can be localised on
an individual ion or spread over
several ions (itinerancy) which can
change with charge state;
relativistic effects are very important
to actinide electronic properties.
This affects the energetic ordering
of the orbitals and means that
spin-orbit coupling should be
included in the calculations since
this will have a large effect on the
spin properties of the actinide atoms
or ions and magnetic properties of
the materials they constitute;
many computer programs cannot
handle f orbitals.
A purely ab initio or quantum mechanical
treatment of these systems is impractical.
The accurate calculation of diffusion
coefficients and solubility curves of gas
atoms and molecules in ceramics
requires models for the ceramic that
contain of the order of 1000 atoms;
more if the oxide does not have a
simple crystal structure or has a
stoichiometric complexity. Thus, a
calculation of the diffusion coefficient
of a hydrogen atom in UO
2
would
require a simulation cell that contained
of the order of 300 uranium ions
(27,000 electrons), which far exceeds
what is technically possible on today’s
super-computers. In fact, it is unlikely
that such calculations will be feasible
within the next ten years, even if
computer power increases at the rapid
pace it has in the past 20 years.
Molecular Modelling
Calculations
Fortunately, one can simplify the
problem of calculating diffusion
coefficients and solubility curves for
atoms and molecules in materials by
setting up models of the system in
which the atoms and ions are
represented by point charges which
interact with each other according to a
set of functions or tabulated values.
This approach, sometimes called
molecular modelling or atomistic
simulation, requires parameters for the
charge on each ion, or pair of charges
for each ion in the shell model (Figure 1).
The charge in the rigid-ion model is
normally set to the formal charge on
the ion, but this is not always the case,
and using partial charges on ions can
lead to unpredictable results and is
another variable that needs to be
defined in the model. The interactions
between the ions and atoms in the
system are defined by a set of functions
representing the potential energy for
each combination of ions and atoms
at different separations; known as
pair-potentials since they act upon
pairs of ions or atoms (Figure 2). These
pair-potentials represent the van der
Waals interactions between species and
when summed with the coulombic
interactions between charged species
the energy of a particular configuration
of the ions is calculable. In systems
which contain highly charged species,
the van der Waals interactions
represented by the pair-potentials
detailed above, contribute approximately
10% of the interaction energy of the
system. Thus, small errors in the
pair-potentials in these systems can
easily be accommodated and will not
be significantly detrimental to the quality
of the results
2
.
Atomistic simulations include interactions
between the electrons and nuclei
implicitly and not explicitly and are
consequently thousands of times
quicker to perform on a system with an
equivalent number of atoms. Key to
the success of atomistic simulations
is the quality of the models for the
individual ions (shell model or better,
for example) and the pair-potentials
between the ions, though in highly
charged systems the pair-potentials
are of lesser importance. In systems
containing uncharged atoms, such as
hydrogen atoms, there are no permanent
coulombic interactions, only instantaneous
ones from temporary polarisation of the
atoms by charged ions in the vicinity of
the atom. In these cases the success of
the simulations will rely heavily upon
the quality of the pair-potentials used
to describe the van der Waals interactions
between the atoms and ions and on the
shell-model parameters for the atoms
themselves
Atomistic simulations can be
parameterised by a number of methods:
by fully quantum mechanical calculations
of isolated systems such as pairs of
ions or atoms; by semi-empirical
20
calculations, such as electron-gas
calculations, on isolated systems
containing pairs of ions, where the
electron density distribution of the ions
is fixed at some unperturbed value and
by fitting to bulk properties of the
material to be simulated, such as bulk
modulus elastic constants and
lattice energy. In systems such as
metal oxides the latter method is most
desirable since it guarantees that the
parameter set will reproduce the bulk
properties of the material and thus its
response to external stimuli such as
isotropic and non-isotropic pressure
changes and heating. In systems
where no crystal structure data can
exist, such as hydrogen interactions
with an actinide oxide, the former two
methods must be applied to the problem.
Molecular modelling has been used to
compute the energy of systems with
the ions being described by one or
other of the two models outlined
above. The Force-Field method relies
on the separation of energy terms into
component parts, each of which can be
equated to a physically meaningful
quantity (equation 1).
In the predominantly ionic systems
under investigation in this work, the
bond stretch term, E
bond
, the bond
rotation term, E
rotation
, and the bond
torsion term, E
torsion
, can be neglected
and only the coulombic term, E
coulombic
,
and van der Waals term, E
vdW
, need be
considered. The coulombic term takes
the form of the function that describes
the potential energy between two
charged particles, but summed over all
possible pair interactions (equation 2).
This coulombic term converges only
very slowly as the separation between
ions increases, so a mathematical
Rigid-Ion Model
±
q
Core shell spring
Shell model q
core
q
shell
Figure 1
a) The rigid-ion model for ions and atoms, and b) the shell model for atoms
and ions. In the rigid-ion model the charge on the point charge representing
the ion is normally the formal charge on the ion, but some parameter sets
do vary.
b) In the shell model, the formal charge on the ion is usually the sum of
the charge on the core and shell, q
ion
= q
core
+ q
shell
Equation 1
) (
bond H vdW torsion rotation bond coulombic total
E E E E E E E
−
+ + + + + =
Equation 2
E
q q e
r
coulombic
i j
o ij
i j
N
j
N

|
(
'
`
J
J
>
−
∑ ∑
2
1
1
4πε
Equation 3
where: N=total number of ions
q
i
, q
j
= formal charge on ion x
r
ij
= separation between ions i and j
∑∑
−
= > ]
]
]
]




− J
J
`
'
(
|
−

1
1
6
exp
N
j
N
j i ij
ij
vdW
r
B
r
A E
ρ
Discovery • The Science & Technology Journal of AWE
21
method known as the Ewald summation
is used to model the interactions. This
requires three terms: a real space
cut-off distance; a reciprocal space cut-off
and an Ewald sum constant. The
mathematical formalism can be found
in standard textbooks
3
.
The van der Waals term can take one of
several forms, but the commonest of
these is the Buckingham potential shown
in equation 3.
These functions produce a potential
energy curve somewhat like that plotted
in Figure 2, having a potential minimum
of depth, D
φ
, at some separation of
ions, R
φ
. The repulsive part of the curve
comes from the exponential term in the
Buckingham function, and emanates
from the repulsion of electron gas
clouds around ions as they are brought
together. The attractive part of the
curve derives from the r
-6
term and is
a response to the dispersion forces
sometimes called van der Waals attraction.
Many properties that we wish to calculate
depend up time and temperature, but
molecular modelling as described
above only deals with static systems.
In order to calculate dynamic properties,
molecular dynamics simulations must
be carried out. These simulations
involve solving the Newtonian equations
of motion at very small time intervals
(time steps) to calculate the positions
of the atoms in a system as a series of
‘snapshots’ in time. Infinite systems
such as crystals can be examined by
using periodic boundary conditions, in
which the simulation cell is repeated on
all sides so that atoms leaving the cell
on one side are never lost to the system
because they re-enter the cell from the
opposite side (Figure 3). The average
Figure 2
Figure 3
D
φ
-1.0
0.0
1.0
2.0
3.0
4.0
5.0
0.0
2.0
4.0 6.0 8.0
10.0
Potential Energy (kJmol
-1
)
Atomic Separation (Å) R
φ
A plot of the Buckingham potential
The periodic boundary condition as applied to a simulation of UO
2-x
. The
simulation cell is surrounded by replicas of itself such that an oxide ion leaving
the cell on the left re-enters the central cell from the right hand cell

22
kinetic energy that the atoms in the
system possess is determined by the
temperature at which the simulation is
carried out.
Molecular dynamics simulations allow
us to calculate properties of materials
such as diffusion coefficients, melting
temperatures and vibrational spectra.
Example of the
Problems Addressed
The reaction of hydrogen with uranium
is a complex process that depends on
many variables, one of which is the
oxide over-layer
1
. The oxide that is
formed on uranium adopts the fluorite
structure which has an empty space at
its centre, known as an interstitial site
(marked by a green X in Figure 4).
Though it is known that the thickness
of the surface oxide layer influences the
rate of reaction of hydrogen with uranium,
it is not known what hydrogen species
is transported through the oxide layer
or the mechanism by which it diffuses.
It is important to understand these
factors in order to quantify the reaction
of hydrogen with uranium and we are
addressing these questions with our
molecular modelling studies.
We are investigating whether the hydrogen
diffuses via interstitial sites, vacancies
in the crystal or grain-boundaries of the
oxide. We have carried out molecular
dynamics simulations at a number of
temperatures of perfect crystals of UO
2
with different species of hydrogen in
order to calculate the propensity of
hydrogen to diffuse via the interstitial
sites within the crystal (Figure 5).
We have built models of imperfect
crystals of UO
2
containing oxide vacancies
Figure 4
Figure 5
The unit cell of UO
2
, with uranium ions represented by blue spheres, oxide ions represented
by red spheres and the interstitial site indicated by a green cross
X
Four unit cells of UO
2
, with uranium ions represented by blue spheres and oxide ions
represented by red spheres. Atom diffusion via interstitial sites as indicated by
curved arrow
Discovery • The Science & Technology Journal of AWE
23
Figure 6
and hydrogen species and calculated
the diffusion coefficients using molecular
dynamics simulations at a range of
temperatures (Figure 6).
The lowest energy structures of 20
low Millar-index surfaces have been
determined by molecular dynamics
simulations at 300 K and this has
revealed that some surfaces are very
stable while some are very unstable as
created. Those that are the most stable
generally require little movement of the
atoms in the top five surface layers and
thus do not change much in appearance.
One such example of a naturally stable
surface is the [1 1 1] surface shown in
Figure 7. Higher energy surfaces, such
as the [4 2 0] shown in Figure 8, tend
to reconstruct to minimise the surface
energy. The reconstruction of the high
energy surfaces can lead to the formation
of microfacets, which are ‘mini-surfaces’
on top of the main surface. In Figure 8
the [4 2 0] surface has reconstructed to
form [1 1 0] and [1 0 0] ‘mini-surfaces’
on top leading to the saw-tooth effect
observed in this case.
A number of models of surfaces have
been combined to create models for
grain-boundaries in UO
2
and used in
molecular dynamics simulations at 300K
to determine which are energetically
the most stable and might provide
pathways for hydrogen diffusion (Figure 9).
Recently we have carried out molecular
dynamics simulations of helium atoms in
our models for UO
2
grain-boundaries
and determined activation barriers
from Arrhenius plots of the diffusion
coefficients at different temperatures
(Figure 10). These simulations allow us to
place an upper limit of the rates of diffusion
of hydrogen at different temperatures.
Four unit cells of UO
2
, with uranium ions represented by blue spheres and
oxide ions represented by red spheres. An arrow indicates the path of a
diffusing atom through an oxide vacancy site (indicated by a green dot)
The calculated lowest energy structure of the [1 1 1] surface of UO
2
at 300K
with the surface running horizontally perpendicular to the page
Figure 7

24
Figure 8
Parameterisation of
Materials
The physical properties of actinide
materials define their response to
external stresses and strains imposed
by the environment and intrinsic
properties of the materials such as
self-diffusion, defect energies and
specific heat capacity. The oxides,
carbides and hydrides of plutonium and
the hydrides of uranium fall into this
category. Little experimental data are
available for these materials since they
are difficult to prepare pure, either in a
given stoichiometry or as a particular
allotrope, are difficult to handle, and, of
course, they are radioactive. Molecular
modelling studies of these materials
require physical properties to
parameterise the models, but these
data are unavailable for the reasons
stated. Thus, they are ideal candidates
for pioneering ab initio computational
studies of their electronic, structural
and mechanical properties.
Workers at Reading University
4
are
using plane-wave density functional
theory to calculate the ground-state
structures of plutonium oxides such as
PuO
2
, α-Pu
2
O
3
and β-Pu
2
O
3
, and from
these their mechanical properties such
as bulk modulus and elastic constants,
C
11
, C
12
and C
44
. Very high cut-off
energies are required in simulations of
these systems, which are equivalent to
extremely high quality basis sets in
traditional ab initio calculations. This
requirement for high cut-off energies
means that these calculations are
extraordinarily computer resource
intensive and so Reading University
makes use of the super-computer
power at AWE. (See Box 1).
X X X
The calculated lowest energy structure of the [4 2 0] surface of UO
2
at 300K
exhibiting the rumpling of microfaceting while the surface runs horizontally
perpendicular to the page
A model of the [1 1 0]-[2 2 1] grain-boundary of UO
2
, with uranium ions
represented by blue spheres and oxide ions represented by red spheres.
The grain-boundary runs horizontally across the page through the centre
of the picture. Three green crosses indicate channels running into the
page, down which diffusion can occur
Figure 9

Discovery • The Science & Technology Journal of AWE
25
In Conclusion
Molecular modelling allows us to probe
materials that are difficult or dangerous
to handle and assess conditions that
are impossible to achieve in the
laboratory. We routinely carry out
simulations of radioactive materials
such as plutonium and uranium
oxides in order to determine diffusion
coefficients of atoms, ions and molecules
at temperatures up to 3000 K, which
would be incredibly difficult to achieve
in the laboratory. Our simulations
are, however, very demanding of
computational resources and require
the huge memory capacity and processor
speed of the AWE super-computers to
complete successfully. Not only are our
calculations computer-intensive, but
0.6
0.8 1.0 1.2 1.4 1.6 1.8
-25.0
-24.5
-24.0
-23.5
-23.0
-22.5
-22.0
-21.5
-21.0
Activation Energy = 25.9 kJmol
-1
+/- 2.5 kJmol
-1
l
n

D
(
D

i
n

m
2
s
-
1
)
1
/
T

(
1
0
-
3
K
)
He diffusion coefficients
An Arrhenius plot of our helium diffusion data for the [3 2 1]-[2 2 1] grain-boundary in UO
2
Figure 10
they test current methodology to its
limits and we require state-of-the-art
computer software to carry out our
simulations.
Our initial results suggest that both
molecular and atomic hydrogen are
immobilised in perfect crystals of UO
2
,
being trapped in the interstitial (vacant)
sites of the lattice. Our current studies,
however, with imperfect crystals,
surfaces and grain-boundaries of UO
2
indicate that helium atoms are much
more mobile in these non-ideal
environments, with activation barriers
to diffusion in the range 20-80 kJmol
-1
.
We are currently defining a parameter
set for hydrogen interactions with
actinide oxides using high level
quantum mechanical methods and will
use this parameter set in our future
simulations of hydrogen solubility and
diffusion in imperfect crystals and
grain-boundaries of uranium and
plutonium oxides. These simulations
will contribute to both a better
understanding of the mechanism of
hydrogen transport through actinide
oxides and data for numerical models
of uranium and plutonium hydriding.
With every simulation, we are gaining
more information about materials that
are technically very challenging to
experimentalists and theoreticians alike,
actinide oxides.

26
The 3 TeraFLOPS IBM super-computer at AWE
Box 1
Figure 12
The beowolf super-computer at AWE
Computer Resources
at AWE
At AWE we have two super-computers
which are constantly upgraded. A
massively parallel IBM SP3 super
computer
5
shown in Figure 11 is equipped
with 1920 Power3 II processors, 2
Terabytes (2 million Megabytes) of
memory, and connected by Nighthawk II
network switches provides 3 TeraFLOPS
(3 x 10
12
floating point operations per
second) of computing power. This
machine is used for calculations that
can be run in parallel and require a lot
of memory. Plane-wave DFT simulations
of actinide oxides typically run on 40 to
250 processors over a period of 2 to
14 days and require 20 to 120 Gigabytes
of memory!
AWE’s beowolf super-computer shown
in Figure 12 has recently been upgraded
to 2.8 GHz Pentium IV Xeon CPUs
delivering 1.2 TeraFLOPS with Gigabit
and Myrinet networking, 552 Gigabytes
of memory and 8.4 Terabytes of disk
space. This machine is used for serial
and parallel molecular dynamics
calculations up to 32 processors requiring
days or weeks of computational time.
Figure 11

Discovery • The Science & Technology Journal of AWE
27
David can be contacted on
e-mail: [email protected]
Author Profile
David Price
David was educated at Kingshurst
Comprehensive then Solihull Sixth
Form College before going up to
Keble College Oxford to read for a
BA(Hons) in Chemistry. He stayed
at Keble to read for a DPhil in
organometallic chemistry, which
encompassed synthetic, mechanistic
and molecular modelling work.
After post-doctoral spells in
Strasbourg and Munich, David
became a research fellow at the
University of Reading funded by
AWE. David joined the materials
modelling team at AWE in 2002
after spending eight years at
Reading University. He currently
works on the ageing of materials,
is a published author and is a full
Member of the Royal Society of
Chemistry and Chartered Chemist.
Acknowledgements
The author wishes to thank Dr. Joseph
Glascott, AWE, Prof. Michael G.B. Drew
and Dr. Mark T. Storr, University of Reading
and the AWE super-computing staff of
Ash Vadgamma, John Dolphin, Mark
Roberts, Paul Tomlinson and Pete Brown.
References
1 J. Glascott, “Hydrogen and Uranium”,
Discovery, Issue 6, January 2003
2 R. W. Grimes, Imperial College,
London University, personal
communication
3 M. P. Allen and D. J. Tildesley,
“Computer Simulations of Liquids”,
Clarendon Press, Oxford
4 M. G. B. Drew, M. T. Storr and
D. W. Price, unpublished work.
5 A. Baxter, “New Supercomputer”,
Discovery, Issue 6, January 2003;
J. Taylor, “History of Computing”,
Discovery, Issue 3, July 2001