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Properties of nanoparticles are influenced by various parameters like size, shape or composition. Comprehensive high throughput characterization techniques are urgently needed to improve synthesis, scale up to production and make way for new applications of multidimensional particulate systems. In this study, we present a method for measuring two-dimensional size distributions of plasmonic nanorods in a single experiment. Analytical ultracentrifuge equipped with a multiwavelength extinction detector is used to record the optical and sedimentation properties of gold nanorods simultaneously. A combination of sedimentation and extinction properties, both depending on diameter and length of the dispersed nanorods, is used to measure two-dimensional distributions of gold nanorod samples. The length, diameter, aspect ratio, volume, surface and cross-sectional distributions can be readily obtained from these results. As the technique can be extended to other non-spherical plasmonic particles and can be used for determining relative amounts of particles of different shapes it provides complete and quantitative insights into particulate systems.

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This paper presents a novel method for the solution of a particular class of structural optimzation problems: the continuous stochastic gradient method (CSG). In the simplest case, we assume that the objective function is given as an integral of a desired property over a continuous parameter set. The application of a quadrature rule for the approximation of the integral can give rise to artificial and undesired local minima. However, the CSG method does not rely on an approximation of the integral, instead utilizing gradient approximations from previous iterations in an optimal way. Although the CSG method does not require more than the solution of one state problem (of infinitely many) per optimization iteration, it is possible to prove in a mathematically rigorous way that the function value as well as the full gradient of the objective can be approximated with arbitrary precision in the course of the optimization process. Moreover, numerical experiments for a linear elastic problem with infinitely many load cases are described. For the chosen example, the CSG method proves to be clearly superior compared to the classic stochastic gradient (SG) and the stochastic average gradient (SAG) method.