Controlling the interfacial and bulk concentrations of spontaneously charged colloids in non-polar media

Stabilization and dispersion of electrical charge by colloids in non-polar media, such as nano-particles or inverse micelles, is significant for a variety of chemical and technological applications, ranging from drug delivery to e-ink. Many applications require knowledge about concentrations near the solid|liquid interface and the bulk, particularly in media where colloids exhibit spontaneous charging properties. By modification of the mean field equations to include the finite size effects that are typical in concentrated electrolytes along with disproportionation kinetics, and by considering high potentials, it is possible to evaluate the width of the condensed double layers near planar electrodes and the bulk concentrations of colloids at steady state. These quantities also provide an estimate of the minimum initial colloid concentration that is required to support electroneutrality in the dispersion bulk, and thus provide insights into the quasi-steady state currents that have been observed in inverse micellar media.Comment: 13 pages, 5 figure

Stationary peaks in a multivariable reaction--diffusion system: Foliated snaking due to subcritical Turing instability

An activator-inhibitor-substrate model of side-branching used in the context of pulmonary vascular and lung development is considered on the supposition that spatially localized concentrations of the activator trigger local side-branching. The model consists of four coupled reaction-diffusion equations and its steady localized solutions therefore obey an eight-dimensional spatial dynamical system in one dimension (1D). Stationary localized structures within the model are found to be associated with a subcritical Turing instability and organized within a distinct type of foliated snaking bifurcation structure. This behavior is in turn associated with the presence of an exchange point in parameter space at which the complex leading spatial eigenvalues of the uniform concentration state are overtaken by a pair of real eigenvalues; this point plays the role of a Belyakov-Devaney point in this system. The primary foliated snaking structure consists of periodic spike or peak trains with $N$ identical equidistant peaks, $N=1,2,\dots \,$, together with cross-links consisting of nonidentical, nonequidistant peaks. The structure is complicated by a multitude of multipulse states, some of which are also computed, and spans the parameter range from the primary Turing bifurcation all the way to the fold of the $N=1$ state. These states form a complex template from which localized physical structures develop in the transverse direction in 2D.Comment: 30 pages, 14 figure

Front propagation and global bifurcations in a multi-variable reaction-diffusion model

We study the existence and stability of propagating fronts in Meinhardt's reaction-diffusion model of branching in one spatial dimension. We identify a saddle-node-infinite-period (SNIPER) bifurcation of fronts that leads to episodic front propagation in the parameter region below propagation failure and show that this state is stable. Stable constant speed fronts exist only above this parameter value. We use numerical continuation to show that propagation failure is a consequence of the presence of a T-point corresponding to the formation of a heteroclinic cycle in a spatial dynamics description. Additional T-points are identified that are responsible for a large multiplicity of different traveling front-peak states. The results indicate that multivariable models may support new types of behavior that are absent from typical two-variable models but may nevertheless be important in developmental processes such as branching and somitogenesis.Comment: 18 pages, 12 figure