Molecular correlations and solvation in simple fluids

We study the molecular correlations in a lattice model of a solution of a low-solubility solute, with emphasis on how the thermodynamics is reflected in the correlation functions. The model is treated in Bethe-Guggenheim approximation, which is exact on a Bethe lattice (Cayley tree). The solution properties are obtained in the limit of infinite dilution of the solute. With $h_{11}(r)$, $h_{12}(r)$, and $h_{22}(r)$ the three pair correlation functions as functions of the separation $r$ (subscripts 1 and 2 referring to solvent and solute, respectively), we find for $r \geq 2$ lattice steps that $h_{22}(r)/h_{12}(r) \equiv h_{12}(r)/h_{11}(r)$. This illustrates a general theorem that holds in the asymptotic limit of infinite $r$. The three correlation functions share a common exponential decay length (correlation length), but when the solubility of the solute is low the amplitude of the decay of $h_{22}(r)$ is much greater than that of $h_{12}(r)$, which in turn is much greater than that of $h_{11}(r)$. As a consequence the amplitude of the decay of $h_{22}(r)$ is enormously greater than that of $h_{11}(r)$. The effective solute-solute attraction then remains discernible at distances at which the solvent molecules are essentially no longer correlated, as found in similar circumstances in an earlier model. The second osmotic virial coefficient is large and negative, as expected. We find that the solvent-mediated part $W(r)$ of the potential of mean force between solutes, evaluated at contact, $r=1$, is related in this model to the Gibbs free energy of solvation at fixed pressure, $\Delta G_p^*$, by $(Z/2) W(1) + \Delta G_p^* \equiv p v_0$, where $Z$ is the coordination number of the lattice, $p$ the pressure, and $v_0$ the volume of the cell associated with each lattice site. A large, positive $\Delta G_p^*$ associated with the low solubility is thus reflected in a strong attraction (large negative $W$ at contact), which is the major contributor to the second osmotic virial coefficient. In this model, the low solubility (large positive $\Delta G_p^*$) is due partly to an unfavorable enthalpy of solvation and partly to an unfavorable solvation entropy, unlike in the hydrophobic effect, where the enthalpy of solvation itself favors high solubility, but is overweighed by the unfavorable solvation entropy.Comment: 9 pages, 2 figure

The solar siblings in the Gaia era

We perform realistic simulations of the Sun's birth cluster in order to predict the current distribution of solar siblings in the Galaxy. We study the possibility of finding the solar siblings in the Gaia catalogue by using only positional and kinematic information. We find that the number of solar siblings predicted to be observed by Gaia will be around 100 in the most optimistic case, and that a phase space only search in the Gaia catalogue will be extremely difficult. It is therefore mandatory to combine the chemical tagging technique with phase space selection criteria in order to have any hope of finding the solar siblings.Comment: To be published in the proceedings of the GREAT-ITN conference "The Milky Way Unravelled by Gaia: GREAT Science from the Gaia Data Releases", 1-5 December 2014, University of Barcelona, Spain, EAS Publications Series, eds Nicholas Walton, Francesca Figueras, and Caroline Soubira

Spacetime algebraic skeleton

The cosmological constant Lambda, which has seemingly dominated the primaeval Universe evolution and to which recent data attribute a significant present-time value, is shown to have an algebraic content: it is essentially an eigenvalue of a Casimir invariant of the Lorentz group which acts on every tangent space. This is found in the context of de Sitter spacetimes but, as every spacetime is a 4-manifold with Minkowski tangent spaces, the result suggests the existence of a "skeleton" algebraic structure underlying the geometry of general physical spacetimes. Different spacetimes come from the "fleshening" of that structure by different tetrad fields. Tetrad fields, which provide the interface between spacetime proper and its tangent spaces, exhibit to the most the fundamental role of the Lorentz group in Riemannian spacetimes, a role which is obscured in the more usual metric formalism.Comment: 13 page