We give two general convergence proofs for random search algorithms. We review the literature and show how our results extend those available for specific variants of the conceptual algorithm studied here. We then exploit the convergence results to examine convergence rates and to actually design implementable methods. Finally we report on some computational experience.

The collapse of flexible polyelectrolytes in a solution of multivalent counterions is studied by means of a two state model. The states correspond to rod-like and spherically collapsed conformations respectively. We focus on the very dilute monomer concentration regime where the collapse transition is found to occur when the charge of the multivalent salt is comparable (but smaller) to that of the monomers. The main contribution to the free energy of the collapsed conformation is linear in the number of monomers N , since the internal state of the collapsed polymer approaches that of an amorphous ionic solid. The free energy of the rod-like state grows as N ln N , due to the electrostatic energy associated with that shape. We show that practically all multivalent counterions added to the system are condensed into the polymer chain, even before the collapse.

We study the attractive interactions between rod-like charged polymers in solution that appear in the presence of multi-valence counterions. The counterions condensed to the rods exhibit both a strong transversal polarization and a longitudinal crystalline arrangement. At short distances between the rods, the fraction of condensed counterions increases, and the majority of these occupy the region between the rods, where they minimize their repulsive interactions by arranging themselves into packing structures. The attractive interaction is strongest for multivalent counterions. Our model takes into account the hard-core volume of the condensed counterions and their angular distribution around the rods. The hard core constraint strongly suppresses longitudinal charge fluctuations.PACS numbers: 61.20. Qg, 61.25.Hq, 87.15.Aa Strongly charged polymers precipitate from a dilute solution into compact structures when high-valence counterions (oppositely charged particles) are added to the solution [1][2][3][4][5][6]. The counterions experience strong electrostatic attractions to the backbone of the chains, and a finite fraction of them "condense", i.e., are found within short distance from the chains [7]. Counterions are more attracted to compact chains or aggregates of rod-like chains. This creates the possibility of a transition from single chains with a small number of condensed counterions to almost neutral aggregates of chains, or even mono-molecular collapse in the case of flexible polymers. These aggregates are stable only when the internal arrangement of the counterions within them, provides a strong enough cohesive energy.In this letter we study the attraction between two rodlike polyelectrolytes. We show that it is essential to include the size and angular degrees of freedom (around the rods) of the counterions as well as the discrete nature of the charge along the polyelectrolytes to find the origin and strength of the counterion mediated attraction. Our work suggests that these factors are also crucial in determining the collapse of flexible and semi-flexible polyelectrolytes recently studied in Refs. [8][9][10][11][12]. Experimental observations show that the size of the precipitating particles is indeed a relevant parameter in the problem [1,6].It has been argued that longitudinal charge fluctuations resulting from the thermal motion of point counterions induce attractions between rod-like polyelelctrolytes [13,14] and induces "buckling" of semiflexible polyelelctrolytes [11,12]. Here, we show that such charge fluctuation are suppressed when the hard core volume of monomers and counterions are taken into account. Instead, we find that the counterions arrangement around the rods create a non-zero transversal polarization as the distance between the chains decreases. At very short distances between the rods we find strong longitudinal correlations but only at very short wavelengths, implying a crystalline state along the rod, reinforcing, for the case of multivalent counterions, the attractive interactio...

Micelles, vesicles, and films composed of two species of incompatible heterogeneous molecules exhibit full internal segregation of the component species. This macroscopic segregation can be inhibited by oppositely charging the two different molecular species. The degree of compatibility achieved by the charges leads to either fully homogenous mixtures or to local segregation and the possible formation of regular patterns. We investigate the induction of periodic surface patterns by the presence of opposite charges in flat films and cylindrical micelles. In the strong segregation limit the incompatibility between species can be described by a line tension parameter gamma. The size of the patterns formed is of the order of a characteristic size L approximately (gamma/sigma(2))(1/2), where sigma is the surface charge density. The pattern symmetry on flat surfaces is function only of the fraction of area covered by the components, f: lamellar for 0.34

For charged systems in heterogeneous dielectric media, a key obstacle for molecular dynamics (MD) simulations is the need to solve the Poisson equation in the media. This obstacle can be bypassed using MD methods that treat the local polarization charge density as a dynamic variable, but such approaches require access to a true free energy functional, one that evaluates to the equilibrium electrostatic energy at its minimum. In this Letter, we derive the needed functional. As an application, we develop a Car-Parrinello MD method for the simulation of free charges present near a spherical emulsion droplet separating two immiscible liquids with different dielectric constants. Our results show the presence of nonmonotonic ionic profiles in the dielectric with a lower dielectric constant.