Diffusion broadening in converging differential mobility analyzers

Although Brownian diffusion limits drastically differential mobility analyzer (DMA) resolution in the nanometer size range, its effects can be moderated in supercritical DMAs running laminarly at very high Reynolds numbers Re. Unfortunately the relatively fast boundary layer growth typical of planar or cylindrical DMAs in current use leads to the appearance of turbulence at Re considerably below 105, even for carefully laminarized flows. Motivated by the fact that flow acceleration stabilizes the boundary layer even at Re>108, we explore theoretically the diffusion broadening phenomenon in converging DMAs. The convective diffusive equations are analyzed for two-dimensional and axisymmetric geometries in the limit of high Re, narrow slits, small aerosol to sheath air flow ratios, and negligible space charge. This yields the line-width of monomobile particles as a quadrature involving the flow and electric fields. There is an optimal position of the inlet slit which minimizes diffusive broadening for given outlet slit position, geometry and sheath air flow rate. For a wide class of potential two-dimensional flows the corresponding optimal resolution can be written explicitly, and is independent of geometry. No such general statements hold in axisymmetric designs, where wall convergence tends to decrease resolution. Explicit results for diffusive broadening, optimal slit locations, and resolution cost as a function of outlet to inlet area ratio are obtained for two-dimensional channels with either straight or hyperbolic walls and for axisymmetric flows with either radial flow between parallel electrodes, or a sink flow within conical electrodes.