The double layer structure of two ionic liquids (ILs), 1-butyl-1-methylpyrrolidinium tris(pentafluoroethyl)trifluorophosphate ([Py1,4]FAP) and 1-ethyl-3-methylimidazolium tris(pentafluoroethyl)trifluorophosphate ([EMIm]FAP) at the polarized Au(111) electrode interface is probed using Atomic Force Microscopy force measurements. The force-separation profiles suggest a multilayered morphology is present at the electrified Au(111)−IL interface, with more near surface layers detected at higher potentials. At the (slightly negative) open circuit potential, multiple ion layers are present, and the innermost layer, in contact with the Au(111) surface, is enriched in the cation due to electrostatic adsorption. Upon applying negative electrode potentials (−1.0 V, −2.0 V), stronger IL near surface structure is detected: both the number of ion layers and the force required to rupture these layers increases. Positive electrode potentials (+1.0 V, +2.0 V) also enhance IL near surface structure, but not as much as negative potentials, because surface-adsorbed anions are less effective at templating structure in subsequent layers than cations. This interfacial structure is not consistent with a double layer in the Stern−Gouy−Chapman sense, as there is no diffuse layer. The structure is consistent with a capicitative double-layer model, with a very small separation distance between the planes of charge.

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We propose a general alternating minimization algorithm for nonconvex optimization problems with separable structure and nonconvex coupling between blocks of variables. To fix our ideas, we apply the methodology to the problem of blind ptychographic imaging. Compared to other schemes in the literature, our approach differs in two ways: (i) it is posed within a clear mathematical framework with practically verifiable assumptions, and (ii) under the given assumptions, it is provably convergent to critical points. A numerical comparison of our proposed algorithm with the current state-of-the-art on simulated and experimental data validates our approach and points toward directions for further improvement.

In this paper we present two Douglas-Rachford inspired iteration schemes which can be applied directly to N-set convex feasibility problems in Hilbert space. Our main results are weak convergence of the methods to a point whose nearest point projections onto each of the N sets coincide. For affine subspaces, convergence is in norm. Initial results from numerical experiments, comparing our methods to the classical (product-space) DouglasRachford scheme, are promising.

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