Nanomaterials (NMs) are materials in which the size of at least one dimension is less than 100 nm. Examples include quantum dots, nanoparticles, “Buckminsterfullerene (C60)”, carbon nanotubes, graphene and TiO2 thin films. Many research groups have investigated the properties of NMs, and they have reported that some of them are clearly different to those of the bulk materials, and depend on the size of the NMs. Examples include melting temperatures, phase transition pressures, fluorescence spectra, catalytic properties and magnetic properties. Recently, a high-pressure study of Te nano-cylinders revealed compressibility effects that are different to those observed in bulk-Te. Although this study reported an elevation of phase transition pressure compared to the bulk, the authors did not investigate the structures of the high-pressure phases, and it is unclear whether the incommensurate phase found at high pressures in bulk-Te was observed or not. Indeed, it is completely unknown whether the incommensurate phases observed in a number of elements at high pressure also exist in nanoparticle samples of the same materials. The search for, and study of, such phases forms the subject of this thesis. Initial studies of commercial selenium nanoparticles (nano-Se) revealed that the incommensurate phase of bulk selenium (Se-IV) is also found in nano-Se. The transition pressures in nano-Se are slightly higher than those of bulk-Se. However, the nano-Se samples were subsequently found not to have the sizes, shapes, and properties claimed by the vendor, which was confirmed by transmission and scanning electron microscopy. Further commercial samples of nano-Se and nano-Bi were also found to be of extremely poor quality. It was clear, therefore, that a detailed study of incommensurate phases in NMs would require us to make our own samples. Bismuth nanoparticles (nano-Bi) with dimensions 51(6), 52(15), 92(13), 128(45), and 138(27) nm have been successfully synthesised by the author in collaboration with the Hybrid Nano Collods group at the University of St. Andrews. On compression, the nano-Bi samples were found to have the same order of phases Bi-I, Bi-II, Bi-III, and Bi-V and phase transitions as found in bulk-Bi, but were found to exhibit larger phase coexistence. The phase transition pressures on pressure increase were higher than those of the bulk materials, and the smaller the diameter of nano-Bi, the higher the phase-transition pressure. This behaviour is similar to, but more extreme than, that found in CdSe nanoparticles. The incommensurate Bi-III structure has been found in nano-Bi under increases in pressure. However, the di↵raction patterns from Bi-III contain additional unaccounted-for peaks, and this phase is referred to as complex Bi-III. The Debye- Scherrer rings from complex Bi-III are smooth, and do not exhibit the spottiness observed in the diffraction patterns of Bi-III obtained from bulk-Bi. This enables full Rietveld refinement of Bi-III in the nano-samples. Complex Bi-III exists from 3 GPa up to 30 GPa, compared to the stable range of only 2.7 to 7.7 GPa of Bi-III in the bulk material. While such a large range of pressure enables the structure of nano-Bi-III to be studied over a much wider pressure range than bulk-Bi-III, such studies were hampered by the existence of the unaccounted-for peaks. In order to get clean, single-phase patterns of Bi-III, samples of this phase were first prepared on pressure decrease from the higher-pressure Bi-V phase, before recompressing them. Single-phase samples of Bi-III were obtained and were found to be stable up to 14-18 GPa. However, because of phase coexistence, diffraction peaks from Bi-III were still visible at pressures as high as ~30 GPa, which is ~3 times larger than the upper limit pressure of existence of bulk-Bi-III. On pressure re-increase, nano-Bi-III has a higher bulk modulus than bulk-Bi-III. The bulk modulus was found to be size-dependent as it is higher when size decreases. Moreover, nano-Bi has a smaller value of the incommensurate wave vector, which is almost pressure independent, but is found to be particles size dependent. The incommensurate wave vector thus becomes another of the structural and physical properties of nanomaterials that is found to be sample-size dependent.

In this thesis I have studied some problems of condensed matter at high pressures and temperatures by means of numerical simulations based on Density Functional Theory (DFT). The stability of MgCO3 and CaCO3 carbonates at the Earth's mantle conditions may play an important role in the global carbon cycle through the subduction of the oceanic crust. By performing ab initio electronic structure calculations, we observed a new high-pressure phase transition within the Pmcn structure of CaCO3. This transformation is characterized by the change of the sp-hybridization state of carbon atom and indicates a change to a new crystal-chemical regime. By performing ab initio Molecular Dynamics simulations we show the new phase to be stable at 250 GPa and 1000K. Thus, the formation of sp3 hybridized bonds in carbonates can explain the stability of MaCO3 and CaCO3 at pressures corresponding to the Earth's lower mantle conditions. We have also calculated phase transition sequence in CaCO3, SrCO3 and BaCO3, and have found that, despite the fact that these carbonates are isostructural and undergo the same type of aragonite to post-aragonite transition, their phase transformation sequences are different at high pressures. The continuous improvement of the high-pressure technique led to the discovery of new composite structures at high pressures and complex phases of many elements in the periodic table have been determined as composite host-guest incommensurate structures. We propose a procedure to accurately describe the structural parameters of an incommensurate phase using ab initio methods by approximating it with a set of analogous commensurate supercells and exploiting the fact that the total energy of the system is a function of structural parameters. By applying this method to the Sc-II phase, we have determined the incommensurate ratio, lattice parameters and Wyckoff positions of Sc-II in excellent agreement with the available experimental data. Moreover, we predict the occurrence of an incommensurate high-pressure phase in Ca from first-principle calculations within this approach. The implementation of DFT in modern electronic structure calculation methods proved to be very successful in predicting the physical properties of a solid at low temperature. One can rigorously describe the thermodynamics of a crystal via the collective excitation of the ionic lattice, and the ab initio calculations give an accurate phonon spectra in the quasi-harmonic approximation. Recently an elegant method to calculate phonon spectra at finite temperature in a self-consistent way by using first principles methods has been developed. Within the framework of self-consistent ab initio lattice dynamics approach (SCAILD) it is possible to reproduce the observed stable phonon spectra of high-temperature bcc phase of Ti, Zr and Hf with a good accuracy. We show that this method gives also a good description of the thermodynamics of hcp and bcc phases of Ti, Zr and Hf at high temperatures, and we provide a procedure for the correct estimation of the hcp to bcc phase transition temperature.