The solubility of rhenium in silicate melts: Implications for the geochemical properties of rhenium at high temperatures

The solubility of rhenium (Re) in a haplobasaltic melt (anorthite-diopside eutectic composition) has been experimentally determined using the mechanically assisted equilibration technique at 1400°C as a function of oxygen fugacity (10−12 < fO2 ≤ 10−7 bar), imposed by CO-CO2 gas mixtures. Samples were analysed by laser ablation-inductively coupled plasma-mass spectrometry (LA-ICP-MS). This is a true microanalytical technique, which allows small-scale sample heterogeneity to be detected, while providing a limit of detection of 2 ppb Re. Time-resolved LA-ICP-MS spectra revealed the presence of suboptically sized micronuggets of Re in all samples, which, because they are present at the 0.5 to 10 ppm level, dominate the true solubilities of Re (<1 ppm at the conditions of the experiment) in bulk analyses of the samples. Nevertheless, the micronuggets could be filtered out from the time-resolved spectra to reveal accurate values of the true Re solubility. A number of time series of samples were taken at constant fO2 to demonstrate that the solubilities converge to a constant value. In addition, solubilities were measured after increasing and decreasing the imposed fO2. The results show that Re dissolves in the silicate melt as ReO2 (Re4+) and ReO3 (Re6+) species, with the latter predominating at typical terrestrial upper-mantle oxygen fugacities. The total solubility of Re is described by the following expression (fO2 in bars): [Re/ppb] = 9.7(±1.9) × 109 (fO2) + 4.2 (±0.3) × 1014 (fO2)1.5Assuming an activity coefficient for Re in Fe-rich metal of 1, this gives a value of DRemet/sil of 5 × 1010 at log fO2 = IW-2, appropriate for metal-silicate partitioning in an homogenously accreting Earth. Thus, Re is indeed very highly siderophile, and the mantle’s abundance cannot be explained by homogenous accretion

Deciding definability in FO2(<h,<v) on trees

We provide a decidable characterization of regular forest languages definable in FO2(<h,<v). By FO2(<h,<v) we refer to the two variable fragment of first order logic built from the descendant relation and the following sibling relation. In terms of expressive power it corresponds to a fragment of the navigational core of XPath that contains modalities for going up to some ancestor, down to some descendant, left to some preceding sibling, and right to some following sibling. We also show that our techniques can be applied to other two variable first-order logics having exactly the same vertical modalities as FO2(<h,<v) but having different horizontal modalities

FO2(<,+1,~) on data trees, data tree automata and branching vector addition systems

A data tree is an unranked ordered tree where each node carries a label from a finite alphabet and a datum from some infinite domain. We consider the two variable first order logic FO2(<,+1,~) over data trees. Here +1 refers to the child and the next sibling relations while < refers to the descendant and following sibling relations. Moreover, ~ is a binary predicate testing data equality. We exhibit an automata model, denoted DAD# that is more expressive than FO2(<,+1,~) but such that emptiness of DAD# and satisfiability of FO2(<,+1,~) are inter-reducible. This is proved via a model of counter tree automata, denoted EBVASS, that extends Branching Vector Addition Systems with States (BVASS) with extra features for merging counters. We show that, as decision problems, reachability for EBVASS, satisfiability of FO2(<,+1,~) and emptiness of DAD# are equivalent

Two-variable logics with some betweenness relations: Expressiveness, satisfiability and membership

We study two extensions of FO2[<], first-order logic interpreted in finite words, in which formulas are restricted to use only two variables. We adjoin to this language two-variable atomic formulas that say, "the letter $a$ appears between positions $x$ and $y$" and "the factor $u$ appears between positions $x$ and $y$". These are, in a sense, the simplest properties that are not expressible using only two variables. We present several logics, both first-order and temporal, that have the same expressive power, and find matching lower and upper bounds for the complexity of satisfiability for each of these formulations. We give effective conditions, in terms of the syntactic monoid of a regular language, for a property to be expressible in these logics. This algebraic analysis allows us to prove, among other things, that our new logics have strictly less expressive power than full first-order logic FO[<]. Our proofs required the development of novel techniques concerning factorizations of words

Origin of Fe3+ in Fe-containing, Al-free Mantle Silicate Perovskite

We have studied the ferrous (Fe2+) and ferric (Fe3+) iron concentrations in Al-free Fe containing Mg-silicate perovskite (Mg-Pv) at pressure (P), temperature (T), and oxygen fugacity (fO2) conditions related to the lower mantle using a thermodynamic model based on ab-initio calculations. We consider the oxidation reaction and the charge disproportionation reaction, both of which can produce Fe3+ in Mg-Pv. The model shows qualitatively good agreement with available experimental data on Fe3+/{\Sigma}Fe ({\Sigma}Fe = total Fe in system), spin transitions, and equations of state. We predict that under lower-mantle conditions Fe3+/{\Sigma}Fe determined by the charge disproportionation is estimated to be 0.01-0.07 in Al-free Mg-Pv, suggesting that low Al Mg-Pv in the uppermost pyrolitic mantle (where majoritic garnet contains most of the Al) and in the harzburgitic heterogeneities throughout the lower mantle contains very little Fe3+. We find that the volume reduction by the spin transition of the B-site Fe3+ leads to a minimum Fe3+/{\Sigma}Fe in Mg-Pv at mid-mantle pressures. The model shows that configurational entropy is a key driving force to create Fe3+ and therefore Fe3+ content is highly temperature sensitive. The temperature sensitivity may lead to a maximum Fe3+/{\Sigma}Fe in Mg-Pv in warm regions at the core-mantle boundary region, such as Large Low Shear Velocity Provinces (LLSVPs), potentially altering the physical (e.g., bulk modulus) and transport (e.g., thermal and electrical conductivities) properties of the heterogeneities

One Quantifier Alternation in First-Order Logic with Modular Predicates

Adding modular predicates yields a generalization of first-order logic FO over words. The expressive power of FO[<,MOD] with order comparison $x<y$ and predicates for $x \equiv i \mod n$ has been investigated by Barrington, Compton, Straubing and Therien. The study of FO[<,MOD]-fragments was initiated by Chaubard, Pin and Straubing. More recently, Dartois and Paperman showed that definability in the two-variable fragment FO2[<,MOD] is decidable. In this paper we continue this line of work. We give an effective algebraic characterization of the word languages in Sigma2[<,MOD]. The fragment Sigma2 consists of first-order formulas in prenex normal form with two blocks of quantifiers starting with an existential block. In addition we show that Delta2[<,MOD], the largest subclass of Sigma2[<,MOD] which is closed under negation, has the same expressive power as two-variable logic FO2[<,MOD]. This generalizes the result FO2[<] = Delta2[<] of Therien and Wilke to modular predicates. As a byproduct, we obtain another decidable characterization of FO2[<,MOD]

Systematic {\em ab initio} study of the phase diagram of epitaxially strained SrTiO3_3

We use density-functional theory with the local-density approximation to study the structural and ferroelectric properties of SrTiO$_3$ under misfit strains. Both the antiferrodistortive (AFD) and ferroelectric (FE) instabilities are considered. The rotation of the oxygen octahedra and the movement of the atoms are fully relaxed within the constraint of a fixed in-plane lattice constant. We find a rich misfit strain-induced phase transition sequence and is obtained only when the AFD distortion is taken into account. We also find that compressive misfit strains induce ferroelectricity in the tetragonal low temperature phase only whilst tensile strains induce ferroelectricity in the orthorhombic phases only. The calculated FE polarization for both the tetragonal and orthorhombic phases increases monotonically with the magnitude of the strains. The AFD rotation angle of the oxygen octahedra in the tetragonal phase increases dramatically as the misfit strain goes from the tensile to compressive strain region whilst it decreases slightly in the orthorhombic (FO4) phase. This reveals why the polarization in the epitaxially strained SrTiO$_3$ would be larger when the tensile strain is applied, since the AFD distortion is found to reduce the FE instability and even to completely suppress it in the small strain region. Finally, our analysis of the average polar distortion and the charge density distribution suggests that both the Ti-O and Sr-O layers contribute significantly to the FE polarization