Bipolar electrochemical mechanism for the propulsion of catalytic nanomotors in hydrogen peroxide solutions

Bimetallic nanorods are propelled in aqueous solutions by the catalytic decomposition of hydrogen peroxide to oxygen and water. Several mechanisms (interfacial tension gradients, bubble recoil, viscous Brownian ratchet, self-electrophoresis) have been proposed for the transduction of chemical to mechanical energy in this system. From Tafel plots of anodic and cathodic hydrogen peroxide reactions at various metal (Au, Pt, Rh, Ni, Ru, and Pd) ultramicroelectrodes, we determine the potential at which the anodic and cathodic reaction rates are equal for each metal. These measurements allow one to predict the direction of motion of all possible bimetallic combinations according to the bipolar electrochemical (or self-electrophoretic) mechanism. These predictions are consistent with the observed direction of motion in all cases studied, providing strong support for the mechanism. We also find that segmented nanorods with one Au end and one poly(pyrrole) end containing catalase, an enzyme that decomposes hydrogen peroxide nonelectrochemically, perform the overall catalytic reaction at a rate similar to that of nanorods containing Au and Pt segments. However, in this case there is no observed axial movement, again supporting the bipolar electrochemical propulsion mechanism for bimetallic nanorods.     

Infinite combinatorics and the foundations of regular variation

The theory of regular variation is largely complete in one dimension, but is developed under regularity or smoothness assumptions. For functions of a real variable, Lebesgue measurability suffices, and so does having the property of Baire. We find here that the preceding two properties have common combinatorial generalizations, exemplified by ‘containment up to translation of subsequences’. All of our combinatorial regularity properties are equivalent to the uniform convergence property.     

The Index Theorem of topological regular variation and its applications

We develop further the topological theory of regular variation of [N.H. Bingham, A.J. Ostaszewski, Topological regular variation: I. Slow variation, LSE-CDAM-2008-11]. There we established the uniform convergence theorem (UCT) in the setting of topological dynamics (i.e. with a group T acting on a homogenous space X), thereby unifying and extending the multivariate regular variation literature. Here, working with real-time topological flows on homogeneous spaces, we identify an index of regular variation, which in a normed-vector space context may be specified using the Riesz representation theorem, and in a locally compact group setting may be connected with Haar measure.