Continuum mechanics at nanoscale. A tool to study trees' watering and recovery

The cohesion-tension theory expounds the crude sap ascent thanks to the negative pressure generated by evaporation of water from leaves. Nevertheless, trees pose multiple challenges and seem to live in unphysical conditions: the negative pressure increases cavitation; it is possible to obtain a water equilibrium between connected parts where one is at a positive pressure and the other one is at negative pressure; no theory is able to satisfactorily account for the refilling of vessels after embolism events. A theoretical form of our paper in the Journal of Theoretical Biology is proposed together with new results: a continuum mechanics model of the disjoining pressure concept refers to the Derjaguin School of physical chemistry. A comparison between liquid behaviour both in tight-filled microtubes and in liquid thin-films is offered when the pressure is negative in liquid bulks and is positive in liquid thin-films and vapour bulks. In embolized xylem microtubes, when the air-vapour pocket pressure is greater than the air-vapour bulk pressure, a refilling flow occurs between the air-vapour domains to empty the air-vapour pockets although the liquid-bulk pressure remains negative. The model has a limit of validity taking the maximal size of trees into account. These results drop inkling that the disjoining pressure is an efficient tool to study biological liquids in contact with substrates at a nanoscale range.Comment: The paper is a review and overlap of my different papers about the watering of trees as a mathematical development of my paper in The Journal of Theoretical Biology. These results are presented together with new researches: transfer of liquid water and vapour between xylem microtubes, an explanation of ultrasounds generated in the watering network considered as sound pipes, numerical calculations of flows in thin liquid films and of Poiseuille flows in xylem microtubes, an estimation of the velocity for the ascent of crude sap and of the recovery time of trees during the spring perio

Motions in liquid-vapour interfaces by using a continuous mechanical model

By using a limit analysis for the motion equations of viscous fluid endowed with internal capillarity, we are able to propose a dynamical expression for the surface tension of moving liquid-vapour interfaces without any phenomenological assumption. The proposed relation extends the static case, yields the Laplace formula in cases of mass transfer across interfacial layers and allows to take the second coefficient of viscosity of compressible fluids into account. We generalize the Maxwell rule in dynamics and directly explain the Marangoni effect.Comment: 15 page

Trimness of Closed Intervals in Cambrian Semilattices

In this article, we give a short algebraic proof that all closed intervals in a $\gamma$-Cambrian semilattice $\mathcal{C}_{\gamma}$ are trim for any Coxeter group $W$ and any Coxeter element $\gamma\in W$. This means that if such an interval has length $k$, then there exists a maximal chain of length $k$ consisting of left-modular elements, and there are precisely $k$ join- and $k$ meet-irreducible elements in this interval. Consequently every graded interval in $\mathcal{C}_{\gamma}$ is distributive. This problem was open for any Coxeter group that is not a Weyl group.Comment: Final version. The contents of this paper were formerly part of my now withdrawn submission arXiv:1312.4449. 12 pages, 3 figure

Electronic correlations, Jahn-Teller distortions and Mott transition to superconductivity in alkali-C60 compounds

The discovery in 1991 of high temperature superconductivity (SC) in A3C60 compounds, where A is an alkali ion, has been rapidly ascribed to a BCS mechanism, in which the pairing is mediated by on ball optical phonon modes. While this has lead to consider that electronic correlations were not important in these compounds, further studies of various AnC60 with n=1, 2, 4 allowed to evidence that their electronic properties cannot be explained by a simple progressive band filling of the C60 six-fold degenerate t1u molecular level. This could only be ascribed to the simultaneous influence of electron correlations and Jahn-Teller Distortions (JTD) of the C60 ball, which energetically favour evenly charged C60 molecules. This is underlined by the recent discovery of two expanded fulleride Cs3C60 isomeric phases which are Mott insulators at ambient pressure. Both phases undergo a pressure induced first order Mott transition to SC with a (p, T) phase diagram displaying a dome shaped SC, a common situation encountered nowadays in correlated electron systems. NMR experiments allowed us to study the magnetic properties of the Mott phases and to evidence clear deviations from BCS expectations near the Mott transition. So, although SC involves an electron-phonon mechanism, the incidence of electron correlations has an importance on the electronic properties, as had been anticipated from DMFT calculations.Comment: Small review article 10 pages, 12 figures. Talk given at the 2011 Eurasia-Pacific Summer School and Conference on Correlated Electrons, in Tutunc (Turkey

Representation of bi-parameter singular integrals by dyadic operators

We prove a dyadic representation theorem for bi-parameter singular integrals. That is, we represent certain bi-parameter operators as rapidly decaying averages of what we call bi-parameter shifts. A new version of the product space T1 theorem is established as a consequence.Comment: 26 page