We study mode-locking in disordered media as a boundary-value problem. Focusing on the simplest class of mode-locking models which consists of a single driven overdamped degree-of-freedom, we develop an analytical method to obtain the shape of the Ar
nold tongues in the regime of low ac-driving amplitude or high ac-driving frequency. The method is exact for a scalloped pinning potential and easily adapted to other pinning potentials. It is complementary to the analysis based on the well-known Shapiros argument that holds in the perturbative regime of large driving amplitudes or low driving frequency, where the effect of pinning is weak.
We construct a mean-field formulation of the thermodynamics of ion solvation in immiscible polar binary mixtures. Assuming an equilibrium planar interface separating two semi-infinite regions of different constant dielectric medium, we study the elec
trostatic phenomenon of differential adsorption of ions at the interface. Using general thermodynamic considerations, we construct the mean-field $Omega$-potential and demonstrate the spontaneous formation of an electric double-layer around the interface necessarily follow. In our framework, we can also relate both the bulk ion densities in the two phases and the distribution potential across the interface to the fundamental Born free energy of ion polarization. We further illustrate this selective ion adsorption phenomenon in respective examples of fully permeable membranes that are neutral, negative, or positive in charge polarity.
We propose a mean-field analytical model to account for the observed asymmetry in the ability to form long-range attraction by the negatively charged colloidal particles and not their equivalently charged positive counterpart. We conjecture that this
asymmetry is due to solvation effects, and we phenomenologically capture its physics by considering the relative strength of this water-induced short-range repulsion between the different charge species. We then apply our model to the colloidal system of negatively charged disks that are neutralized by a sea of counterions and strongly absorbed to an interface in a compressible binary system. We demonstrate the resulting coexistence between a dilute isotropic ionic phase and a condensed hexagonal lattice phase as a function of density and interaction strength.
