Anal. Chem. 1992, 64, 1312
1312
CORRESPONDENCE
Space Charge Effects in Ion Mobility Spectrometry Glenn E. Spangler Environmental Technologies Group, Inc., 1400 Taylor Avenue, Baltimore, Maryland 21284-9840
In a recent review of ion mobility spectrometry, 8t. Louis and Hill concluded, on the basis of negligible repulsion between nearest neighboring ions, that coulombic repulsion contributed insignificantly to the spreading of ion mobility peaks.1 Because more advanced treatments of space charge are much less conclusive, I present here another argument, based on that by R. Johnson, as summarized in Mason and McDaniel's book.2 Consider a slab of ions of finite width extending infinitely into space. Suppose this ion slab is migrating through a drift tube. During the migration, the ions diffuse and the ion number density eventually assumes a Gaussian distribution. The ion concentration at time t at any point z along the drift tube axis is
where no is the initial ion concentration, D is the diffusion coefficient, and Z d is the location for the center of the Gaussian distribution. The root-mean-square displacement of the ions from the center of the distribution is given by
4== ['Jno" --
(2
- Zd)2nd(Z- I d ) ] 1'2 =
a
(2)
If this root-mean-square displacement is denoted by R, R will increase with time as dRldt = DJR. At the same time, the ion cloud will be expanding due to interionic repulsion. From Gauss's law in electrostatics, the electric field, E,, induced by the space charge at the edges of the ion distribution is3
where Ud is the drift velocity for the ions, t , is the "gate-on" (1) St. Louis, R. H.; Hill, H. H., Jr. Crit. Rev. Anal. Chem. 1990, 21 (5),321-355. (2) Mason, E. A,; McDaniel, E. W. Transport Properties of Ions in Gases; Wiley: New York, 1988; p 7.
0003-2700/92/0364-1312$03.00/0
time used to introduce the ions into the drift region at the shutter grid, Udtg is the width of the ion slab, and €0 is the permittivity of free space. The ions exposed to this field will have their drift velocity modified by lvd,l = KE, where K is the ion mobility. If space charge effects are to be negligible, then Iu~sJ
