The Mayer series of the Lennard-Jones gas: improved bounds for the convergence radius

We provide a lower bound for the convergence radius of the Mayer series of the Lennard-Jones gas which strongly improves on the classical bound obtained by Penrose and Ruelle 1963. To obtain this result we use an alternative estimate recently proposed by Morais et al. (J. Stat. Phys. 2014) for a restricted class of stable and tempered pair potentials (namely those which can be written as the sum of a non-negative potential plus an absolutely integrable and stable potential) combined with a method developed by Locatelli and Schoen (J. Glob. Optim. 2002) for establishing a lower bound for the minimal interatomic distance between particles interacting via a Morse potential in a cluster of minimum-energy configurations

On Lennard-Jones type potentials and hard-core potentials with an attractive tail

We revisit an old tree graph formula, namely the Brydges-Federbush tree identity, and use it to get new bounds for the convergence radius of the Mayer series for gases of continuous particles interacting via non absolutely summable pair potentials with an attractive tail including Lennard-Jones type pair potentials

Uncertainty quantification for classical effective potentials

Effective potentials are an essential ingredient of classical molecular dynamics simulations. Little is understood of the errors incurred in representing the complex energy landscape of an atomic configuration by an effective potential containing considerably fewer parameters. This thesis details the introduction of an uncertainty quantification framework into the potential fitting process within the potfit force matching code. The probabilistic sloppy model method has been implemented within potfit as a means to quantify the uncertainties in analytic potential parameters, and in subsequent quantities measured using the fitted potential. Uncertainties in the effective potential are propagated through molecular dynamics simulations to obtain uncertainties in quantities of interest, which are a measure of the confidence in the model predictions. The implementation has been designed to fit flexibly within the existing potfit workflow, and is generalised to work with any potential model or material. The uncertainty quantification software contains a variety of controllable parameters, which provide the user with diagnostic capabilities to understand the nature of the fitting landscape defined by their potential model and reference data. The implementation is available for use by the materials modelling community as part of the open source potfit software. The uncertainty quantification technique is demonstrated using three potentials for nickel: two simple pair potentials, Lennard-Jones and Morse, and a local density dependent EAM potential. A sloppy model fit to ab initio reference data is constructed for each potential to calculate the uncertainties in lattice constants, elastic constants and thermal expansion. These can be used to show the unsuitability of pair potentials for nickel. In contrast, with EAM we observe a decreased uncertainty in the model predictions. This shows that our method can capture the effects of the error incurred in the potential generation process without resorting to comparison with experiment or ab inito calculations, which is an essential part to assess the predictive power of molecular dynamics simulations

Computational modelling of A2BO4 materials for solid oxide fuel cells

Solid oxide fuel cells are a clean, attractive and highly efficient alternative to traditional methods of power generation. However, their high operating temperature prevents their widespread use, as it makes them prohibitively expensive and causes stability issues. In response to this, new materials which exhibit fast oxide ion conduction at low-intermediate temperatures have gained considerable research interest. In this thesis, atomic scale computational modelling techniques have been employed to investigate defects, dopants and oxide ion conductivity in the A2BO4 materials; Cd2GeO4 and Ba2TiO4. Calculations suggest that these materials' parent structures are poor oxide ion conductors due to their highly unfavourable intrinsic defect formation energies of 3.00-10.56 eV defect-1. Results also indicate that oxide ion interstitials can be formed in Cd2GeO4through trivalent doping of the Cd sites. In Ba2TiO4, monovalent and trivalent doping of its Ba and Ti sites respectively induces the formation of oxide ion vacancies. In both materials, strong dopant-oxide ion defect associations are present. Interestingly, only Cd2GeO4 shows enhanced oxide ion migration upon doping, the defects in Ba2TiO4 being effectively immobile. This suggest that the oxide ion vacancies are more intensely associated with their causal dopant ions. With an average migration barrier of ~0.79 eV, oxide ions diffuse in Cd2GeO4 via a “knock-on” mechanism down the a-axis and a stepwise mechanism along the c-axis. Despite this, defect trapping confines the interstitials to the dopant rich regions of the cell, resulting in poor oxide ion diffusion on the order of 1x10-8 cm2 s-1 at 1273 K. Generally, defects are found to be more stable in the α’-phase of Ba2TiO4, suggesting, in agreement with experiment, that they are likely to stabilise the α’-phase at reduced temperatures. Subsequent investigations, also in accord with experiment, reveal carbonate impurities are likely to be common in pristine and doped Ba2TiO4 systems alike, and that their presence will stabilise the α’-phase. The hydroxide type defects formed upon water incorporation are shown to be low in energy in Ba2TiO4 systems containing oxide ion vacancies or interstitials. Both carbonate and hydroxide type defects are shown to bind aggressively to any oxide ion defects present, reducing their mobility

Atomistic modelling of diffusion

This thesis describes the dimer method, which is an algorithm that can be used to find state transitions in an atomistic system, and the application of this method to two different atomistic diffusion problems. The dimer method is an algorithm that locates the saddle points of a potential field of arbitrary dimensionality. These saddle points correspond to the points of transition between metastable states of an atomistic system. A number of improvements to the algorithm of the dimer method have been described and implemented in this work. The first atomistic problem to be described is the diffusion of Au adatoms on a face-centred cubic Au(100) surface. By applying the dimer method to this system, a number of state transitions involving varying numbers of atoms are discovered, from the initial configuration of a single adatom on the surface and from configurations of two adatoms close together. [Continues.