Contractibility of the Space of Opers for Classical Groups

The geometric Langlands program is an exciting direction of research, although a lot of progress has been made there are still open questions and gaps. One of the necessary steps for this program is the proof of the existence and the cohomological triviality (more precisely, the O- contractibility) of the space of rational opers. In this paper, we prove the homotopical contractibility of the space of rational opers for classical groups

Complex Fluids with Mobile Charge-Regulated Macro-Ions

We generalize the concept of charge regulation of ionic solutions, and apply it to complex fluids with mobile macro-ions having internal non-electrostatic degrees of freedom. The suggested framework provides a convenient tool for investigating systems where mobile macro-ions can self-regulate their charge (e.g., proteins). We show that even within a simplified charge-regulation model, the charge dissociation equilibrium results in different and notable properties. Consequences of the charge regulation include a positional dependence of the effective charge of the macro-ions, a non-monotonic dependence of the effective Debye screening length on the concentration of the monovalent salt, a modification of the electric double-layer structure, and buffering by the macro-ions of the background electrolyte.Comment: 7 pages, 5 figure

Charge regulation: a generalized boundary condition?

The three most commonly-used boundary conditions for charged colloidal systems are constant charge (insulator), constant potential (conducting electrode) and charge regulation (ionizable groups at the surface). It is usually believed that the charge regulation is a generalized boundary condition that reduces in some specific limits to either constant charge or constant potential boundary conditions. By computing the disjoining pressure between two symmetric planes for these three boundary conditions, both numerically (for all inter-plate separations) and analytically (for small inter-plate separations), we show that this is not, in general, the case. In fact, the limit of charge regulation is a separate boundary condition, yielding a disjoining pressure with a different characteristic separation-scaling. Our findings are supported by several examples demonstrating that the disjoining pressure at small separations for the charge regulation boundary-condition depends on the details of the dissociation/association process.Comment: 6 pages, 3 figure

Surface Tension of Electrolyte Solutions: A Self-consistent Theory

We study the surface tension of electrolyte solutions at the air/water and oil/water interfaces. Employing field-theoretical methods and considering short-range interactions of anions with the surface, we expand the Helmholtz free energy to first-order in a loop expansion and calculate the excess surface tension. Our approach is self-consistent and yields an analytical prediction that reunites the Onsager-Samaras pioneering result (which does not agree with experimental data), with the ionic specificity of the Hofmeister series. We obtain analytically the surface-tension dependence on the ionic strength, ionic size and ion-surface interaction, and show consequently that the Onsager-Samaras result is consistent with the one-loop correction beyond the mean-field result. Our theory fits well a wide range of concentrations for different salts using one fit parameter, reproducing the reverse Hofmeister series for anions at the air/water and oil/water interfaces.10.1029Comment: 5 pages, 2 figure

Charged Membranes: Poisson-Boltzmann theory, DLVO paradigm and beyond

In this chapter we review the electrostatic properties of charged membranes in aqueous solutions, with or without added salt, employing simple physical models. The equilibrium ionic profiles close to the membrane are governed by the well-known Poisson-Boltzmann (PB) equation. We analyze the effect of different boundary conditions, imposed by the membrane, on the ionic profiles and the corresponding osmotic pressure. The discussion is separated into the single membrane case and that of two interacting membranes. For one membrane setup, we show the different solutions of the PB equation and discuss the interplay between constant-charge and constant-potential boundary conditions. A modification of the PB theory is presented to treat the extremely high counter-ion concentration in the vicinity of a charge membrane. For two equally-charged membranes, we analyze the different pressure regimes for the constant-charge boundary condition, and discuss the difference in the osmotic pressure for various boundary conditions. The non-equal charged membranes is reviewed as well, and the crossover from repulsion to attraction is calculated analytically. We then examine the charge-regulation boundary condition and discuss its effects on the ionic profiles and the osmotic pressure for two equally-charged membranes. In the last section, we briefly review the van der Waals (vdW) interactions and their effect on the free energy between two planar membranes. We explain the simple Hamaker pair-wise summation procedure, and introduce the more rigorous Lifshitz theory. The latter is a key ingredient in the DLVO theory, which combines repulsive electrostatic with attractive vdW interactions, and offers a simple explanation for colloidal or membrane stability. Finally, the chapter ends by a short account of the limitations of the approximations inherent in the PB theory.Comment: 57 pages, 19 figures, From the forthcoming Handbook of Lipid Membranes: Molecular, Functional, and Materials Aspects. Edited by Cyrus Safinya and Joachim Radler, Taylor & Francis/CRC Press, 201