Equations Governing Single and Tandem DMA Configurations and a New Lognormal Approximation to the Transfer Function

Abstract

The basic equations governing the responses of single and tandem differential mobility analyzer (DMA) systems are summarized. Particle diffusion within the DMA resulting in broadening of the transfer function is included in this analysis. For tandem DMA (TDMA) work, a given particle diameter exiting the first DMA is modeled in the following conditioner as growing into a multi-modal lognormal distribution before entering the second DMA. Approximations and solution techniques for both single and tandem DMA systems are discussed. A new lognormal approximation to the DMA transfer function is introduced leading to a simple lognormal form for the TDMA response function. The maximum absolute error of the transfer function is 0.10 at 200 nm in the range plus or minus one and a half standard deviations of the lognormal fit. It is 0.035 at 3 nm in the range plus or minus two standard deviations of the lognormal fit. The maximum fractional error in the calculated TDMA response is no more than 0.08 at 200 nm and 3 nm in the range plus or minus one standard deviation of the lognormal fit.

Acknowledgments

We gratefully acknowledge the assistance of Mr. Chongai Kuang, who proofread the manuscript and checked all equations. This research was supported, in part, by the Office of Science (BER), U.S. Department of Energy, grant DE-FG-02-05ER63997.

Notes

¹ As with the axial flow DMA and L, R _in and R _out are defined according to the change in electric flux function from inlet to outlet. The flux function at each of these is equal to the asymptotic value approached deep within the inlet or outlet. For the inlet R _in is half way between the walls of the inlet slit. For the outlet R _out is on the axis of the exit tube. Hence, R _out = 0.