ION-DROPLET MOBILITY IN CRYOGENIC 4He VAPOR

We report measurements of positive ion mobility in %e vapor at low temperatures and pressures. The results support a droplet model of ionic stucture. The Gibbs-Thompson Equation predicts that a -(c/r6) arising from each of the atoms in the drop. positive ion in helium vapor will condense aboutitAssuming pairwise additivity we can integrate u v self a droplet of radius RT, where over a drop of radius R to obtain the complete po1 2y ae2 (1) tential an(~/~sAT) = {= rcn!L 3R+r) (3R-rh ae2 Here y is the surface tension, a the atomic polariU(r) = IZr GR+r)' (R-r) 2r4 (2) zability, na the liquid density, and (p/pSAT) is the pressure saturation ratio. The calculation uses macroscopic thermodynamics and assumes an ideal vapor. To see if the droplets do indeed form we have measured the positive ion mobility in '~e vapor at low pressures and temperatures where the vapor may be presumed ideal. Since only binary collisions are important, if the vapor atom-ion interaction is pressure independent, the reduced mobility pr 5 (n/nref)u will be pressure independent also. Here 3 n is the vapor density and n E 2.69 x 1019cm . ref An observed decrease in 1-1, with increasing p on the other hand, would indicate an increasing atom-ion interaction and thus a growing droplet. Such behavior is not clearly evident in previous measurements at higher pressures and temperature /I/. An example of our data is given in figure 1. The 30 % decrease in p with increasing p/pSAT provides strong qualitative evidence of droplet growth. The observation of a well defined mobility indicates in addition that formation of droplets is rapid on the time scale of our measurements (% 1 ms) even at pressures of 1 torr or less. In order to obtain a measure of droplet size from mobility measurements one must adopt a model of the interaction between a vapor atom and theiondroplet and then calculate 1 , . We assume that the important interactions are the polarization potential, u = -(ae2/2r4) arising from the central P charge, and the Van der Waals potential u = * Supported by NSF Grant # DMR 76-1 1 1 1 1 Fig. 1 : Experimental reduced mobility vs. (p/pSAT) at T = 1.758 K and pSAT = 10.75 torr. The classical trajectories in such a steep poten.tial are characterized by a rapid transition from small deflections at larger values of the impact parameter b to orbits which spiral into the su'rfaae of the drop at smaller b. Sknce recent surface scattering measurements /2/ indicate that an atom incident on the helium,surface is almost certainly absorbed, it is likely that a similar result occurs at the droplet surface also. It is a simple matter to numerically evaluate the velocity dependent critical impact parameter b (v) below which spiral orbits occur /3/ and we find them to be relatively slowly varying near thermal velocities. It seems then a reasonable first approximation to ignore momentum exchange arising from trajectories with b > b and treat the interaction as a totally inelastic process characterized by a cross section Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1978639 ~b'(v). For a massive ion the mean momentum exchanged in such an inelastic event is &, where m -+ is the atomic mass and v the relative velocity, since captured atoms are reemitted randomly in the ion rest frame. We have calculated the mobility assuming this interaction and obtain a resultwhich is an extension of one reported by Epstein 141. where Eb@ 2 a'$ ~:(vflv~e"'dv (4) and a (m/2kT). Since v5 exp(-av2) is sharply ked about vo Z (5/2~4)'/~ and bs is slowly varying, it is a good approximation to replace b (v) by b (v ) in the integrand and thus obtain b b 0 S (-1. The values of the capture radius bo calculated in this way for the theoretical drop radii are shown in figure 2. Fig. 2 : Theoretical droplet radius %, calculated capture radius bo, and experimental capture radius (EXPTZ vs. (p/pSAT) at T = 1.758 K. We may compare these predictions with experiment by solving eq. (3) to obtain an "experimental" boThe result is shown as the curve labeled "EXPT" in figure 2, and proves to be larger than the theoretical value by about 17 %. There are at least three effects neglected in our simplified' model which would increase the theoretical cross section and improve agreement with experiment. These are curvature dependence of y which modifies eq. (I) and increases RT, diffuseness of the droplet surface, which affects u(r), and momentum exchange arising from elastic scattering with b > bs. In summary, we have shown qualitatively that droplets do indeed form about positive ions in cryogenic '~e vapor, and that they do so rapidly even at low pressures. In addition we have presented an approximate model of the atom-droplet interaction which predicts values of the mobility in reasonable agreement with experiment when the droplet size is calculated using macroscopic thermodynamics. Further refinements are suggested which should improve that agreement.