The earth’s core: an approach from first principles

The Earth’s core is largely composed of iron (Fe), alloyed with less dense elements such as sulphur, silicon and/or oxygen. The phase relations and physical properties of both solid and liquid Fe-alloys are therefore of great geophysical importance. As a result, over the past fifty years the properties of Fe and its alloys have been extensively studied experimentally. However, achieving the extreme pressures (up to 360 GPa) and temperatures (~6000K) found in the core provide a major experimental challenge, and it is not surprising that there are still considerable discrepancies in the results obtained by using different experimental techniques. In the past fifteen years quantum mechanical techniques have been applied to predict the properties of Fe. Here we review the progress that has been made in the use of first principles methods to study Fe and its alloys, and as a result of these studies we conclude: (i) that pure Fe adopts an hexagonal close packed structure under core conditions and melts at ~6200 K at 360 GPa, (ii) that thermodynamic equilibrium and observed seismic data are satisfied by a liquid Fe alloy outer core with a composition of ~10 mole% S (or Si) and 8 mole% O crystallising at ~ 5500 K to give an Fe alloy inner core with ~8 mole% S (or Si) and 0.2 mole % O, and (iii) that with such concentrations of S (or Si), an Fe alloy might adopt a body centred cubic structure in all or part of the inner core. In the future the roles of Ni, C, H and K in the core need to be studied, and techniques to predict the transport and rheological properties of Fe alloys need to be developed

Disorder, pre-stress and non-affinity in polymer 8-chain models

To assess the role of single-chain elasticity, non-affine strain fields and pre-stressed reference states we present and discuss the results of numerical and analytical analyses of modified 8-chain Arruda-Boyce model for cross-linked polymer networks. This class of models has proved highly successful in modeling the finite-strain response of flexible rubbers. We extend it to include the effects of spatial disorder and the associated non-affinity, and use it to assess the validity of replacing the constituent chain's nonlinear elastic response with equivalent linear, Hookean springs. Surprisingly, we find that even in the regime of linear response, the full polymer model gives very different results from its linearized counterpart, even though none of the chains are stretched beyond their linear regime. We demonstrate that this effect is due to the fact that the polymer models are under considerable pre-stress in their ground state. We show that pre-stress strongly suppresses non-affinity in these unit cell models, resulting in a marked stiffening of the bulk response. The effects of pre-stress we discuss may explain why fully affine mechanical models, in many cases, predict the bulk mechanical response of disordered stiff polymer networks so well.Comment: 29 pages, 7 figures. Submitted to J. Mech. Phys. Solid

The Gibbs-Thomson formula at small island sizes - corrections for high vapour densities

In this paper we report simulation studies of equilibrium features, namely circular islands on model surfaces, using Monte-Carlo methods. In particular, we are interested in studying the relationship between the density of vapour around a curved island and its curvature-the Gibbs-Thomson formula. Numerical simulations of a lattice gas model, performed for various sizes of islands, don't fit very well to the Gibbs-Thomson formula. We show how corrections to this form arise at high vapour densities, wherein a knowledge of the exact equation of state (as opposed to the ideal gas approximation) is necessary to predict this relationship. Exploiting a mapping of the lattice gas to the Ising model one can compute the corrections to the Gibbs-Thomson formula using high field series expansions. We also investigate finite size effects on the stability of the islands both theoretically and through simulations. Finally the simulations are used to study the microscopic origins of the Gibbs-Thomson formula. A heuristic argument is suggested in which it is partially attributed to geometric constraints on the island edge.Comment: 27 pages including 7 figures, tarred, gzipped and uuencoded. Prepared using revtex and espf.sty. To appear in Phys. Rev.

Joining Forces of Bayesian and Frequentist Methodology: A Study for Inference in the Presence of Non-Identifiability

Increasingly complex applications involve large datasets in combination with non-linear and high dimensional mathematical models. In this context, statistical inference is a challenging issue that calls for pragmatic approaches that take advantage of both Bayesian and frequentist methods. The elegance of Bayesian methodology is founded in the propagation of information content provided by experimental data and prior assumptions to the posterior probability distribution of model predictions. However, for complex applications experimental data and prior assumptions potentially constrain the posterior probability distribution insufficiently. In these situations Bayesian Markov chain Monte Carlo sampling can be infeasible. From a frequentist point of view insufficient experimental data and prior assumptions can be interpreted as non-identifiability. The profile likelihood approach offers to detect and to resolve non-identifiability by experimental design iteratively. Therefore, it allows one to better constrain the posterior probability distribution until Markov chain Monte Carlo sampling can be used securely. Using an application from cell biology we compare both methods and show that a successive application of both methods facilitates a realistic assessment of uncertainty in model predictions.Comment: Article to appear in Phil. Trans. Roy. Soc.

Calculation of the free energy of crystalline solids

The prediction of the packing of molecules into crystalline phases is a key step in understanding the properties of solids. Of particular interest is the phenomenon of polymorphism, which refers to the ability of one compound to form crystals with different structures, which have identical chemical properties, but whose physical properties may vary tremendously. Consequently the control of the polymorphic behavior of a compound is of scientific interest and also of immense industrial importance. Over the last decades there has been growing interest in the development of crystal structure prediction algorithms as a complement and guide to experimental screenings for polymorphs. The majority of existing crystal structure prediction methodologies is based on the minimization of the static lattice energy. Building on recent advances, such approaches have proved increasingly successful in identifying experimentally observed crystals of organic compounds. However, they do not always predict satisfactorily the relative stability among the many predicted structures they generate. This can partly be attributed to the fact that temperature effects are not accounted for in static calculations. Furthermore, existing approaches are not applicable to enantiotropic crystals, in which relative stability is a function of temperature. In this thesis, a method for the calculation of the free energy of crystals is developed with the aim to address these issues. To ensure reliable predictions, it is essential to adopt highly accurate molecular models and to carry out an exhaustive search for putative structures. In view of these requirements, the harmonic approximation in lattice dynamics offers a good balance between accuracy and efficiency. In the models adopted, the intra-molecular interactions are calculated using quantum mechanical techniques; the electrostatic inter-molecular interactions are modeled using an ab-initio derived multipole expansion; a semi-empirical potential is used for the repulsion/dispersion interactions. Rapidly convergent expressions for the calculation of the conditionally and poorly convergent series that arise in the electrostatic model are derived based on the Ewald summation method. Using the proposed approach, the phonon frequencies of argon are predicted successfully using a simple model. With a more detailed model, the effects of temperature on the predicted lattice energy landscapes of imidazole and tetracyanoethylene are investigated. The experimental structure of imidazole is Abstract | ii correctly predicted to be the most stable structure up to the melting point. The phase transition that has been reported between the two known polymorphs of tetracyanoethylene is also observed computationally. Furthermore, the predicted phonon frequencies of the monoclinic form of tetracyanoethylene are in good agreement with experimental data. The potential to extend the approach to predict the effect of temperature on crystal structure by minimizing the free energy is also investigated in the case of argon, with very encouraging results.Open Acces

Perturbation theory for non-spherical fluids based on discretization of the interactions

7 páginas, 5 figuras; PACS: 65.20.De, 61.20.JaAn extension of the discrete perturbation theory [A. L. Benavides and A. Gil-Villegas, Mol. Phys. 97(12), 1225 (1999)10.1080/00268979909482924] accounting for non-spherical interactions is presented. An analytical expression for the Helmholtz free energy for an equivalent discrete potential is given as a function of density, temperature, and intermolecular parameters with implicit shape dependence. The presented procedure is suitable for the description of the thermodynamics of general intermolecular potential models of arbitrary shape. The overlap and dispersion forces are represented by a discrete potential formed by a sequence of square-well and square-shoulders potentials of shape-dependent widths. By varying the intermolecular parameters through their geometrical dependence, some illustrative cases of square-well spherocylinders and Kihara fluids are considered, and their vapor-liquid phase diagrams are tested against available simulation data. It is found that this theoretical approach is able to reproduce qualitatively and quantitatively well the Monte Carlo data for the selected potentials, except near the critical region.A.L.B. acknowledges funding received by Grant No. 152684 CONACYT (México). F.G. acknowledges funding through Project No. P07-FQM-02600 (Junta de Andalucía-FEDER) for his postdoctoral fellowship.Peer reviewe

Equation of State Based Slip Spring Model for Entangled Polymer Dynamics

A mesoscopic, mixed particle- and field-based Brownian dynamics methodology for the simulation of entangled polymer melts has been developed. Polymeric beads consist of several Kuhn segments, and their motion is dictated by the Helmholtz energy of the sample, which is a sum of the entropic elasticity of chain strands between beads, slip springs, and nonbonded interactions. The entanglement effect is introduced by the slip springs, which are springs connecting either nonsuccessive beads on the same chain or beads on different polymer chains. The terminal positions of slip springs are altered during the simulation through a kinetic Monte Carlo hopping scheme, with rate-controlled creation/destruction processes for the slip springs at chain ends. The rate constants are consistent with the free energy function employed and satisfy microscopic reversibility at equilibrium. The free energy of nonbonded interactions is derived from an appropriate equation of state, and it is computed as a functional of the local density by passing an orthogonal grid through the simulation box; accounting for it is necessary for reproducing the correct compressibility of the polymeric material. Parameters invoked by the mesoscopic model are derived from experimental volumetric and viscosity data or from atomistic molecular dynamics simulations, establishing a "bottom-up" predictive framework for conducting slip spring simulations of polymeric systems of specific chemistry. The mesoscopic simulation methodology is implemented for the case of cis-1,4-polyisoprene, whose structure, dynamics, thermodynamics, and linear rheology in the melt state are quantitatively predicted and validated without a posteriori fitting the results to experimental measurements.Comment: 80 pages, 17 figure

Disjoining pressure of planar adsorbed films

Frumkin-Derjaguin theory of interfacial phase transitions and in particular the concept of the disjoining pressure of a planar adsorbed film is reviewed and then discussed in terms of statistical mechanical formulations of interfacial phase transitions beyond mean-field.Comment: 11 pages including the two figure