We present a general parameter study, in which the abundance of interstellar argonium (ArH$^+$) is predicted using a model for the physics and chemistry of diffuse interstellar gas clouds. Results have been obtained as a function of UV radiation field, cosmic-ray ionization rate, and cloud extinction. No single set of cloud parameters provides an acceptable fit to the typical ArH$^+$, OH$^+$ and $rm H_2O^+$ abundances observed in diffuse clouds within the Galactic disk. Instead, the observed abundances suggest that ArH$^+$ resides primarily in a separate population of small clouds of total visual extinction of at most 0.02 mag per cloud, within which the column-averaged molecular fraction is in the range $10^{-5} - 10^{-2}$, while OH$^+$ and $rm H_2O^+$ reside primarily in somewhat larger clouds with a column-averaged molecular fraction $sim 0.2$. This analysis confirms our previous suggestion that the argonium molecular ion is a unique tracer of almost purely atomic gas.
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