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Submitted on 1 Jan 1978
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ION-DROPLET MOBILITY IN CRYOGENIC 4He
VAPOR
J. Northby, G. Akinci
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, suppliment au no 8, zTome 39, aolit 1978, page C6-84
ION-DROPLET MOBILITY IN CRYOGENIC
4 ~ eVAPOR*
J.A. Northby and G. AkinciPhysics D e p a r h e n t , h i v e r s i t y o f Rhode I s l a n d Kingston, Rhode I s l a n d , 02881 U.S.A.
Rdsumd.- On pr6sente des mesures de mobilitd des ions positifs dans le gaz '~e sous des tempdratures et des pressions basses. Les rdsultats rdalisds paraissent confirmer la thsorie selon laquelle la structure de l'ion ressemblerait 1 celle d'une gouttelette chargse.
Abstract.- 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 aboutit- Assuming pairwise additivity we can integrate u v
self a droplet of radius RT, where over a drop of radius R to obtain the complete po-
1 2y ae2
(1) tential
an(~/~sAT) = {=
-
rcn!L
3R+r) (3R-rh ae2Here y is the surface tension, a the atomic polari- U(r) =
- -
IZr GR+r)' (R-r) 2r4 (2)zability, na the liquid density, and (p/pSAT) is the pressure saturation ratio. The calculation uses ma- croscopic thermodynamics and assumes an ideal vapor. To see if the droplets do indeed form we have mea- sured 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
-
3n 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 beha- vior is not clearly evident in previous measure- ments 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 theion- droplet and then calculate 1-1,. We assume that the important interactions are the polarization poten- tial, 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 1Fig. 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 cri- tical 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 ine- lastic process characterized by a cross section
~b'(v). For a massive ion the mean momentum ex- changed 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
E b @ 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 b0 S
(-1. The values of the capture radius bo cal- culated 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"
bo- The result is shown as the curve labeled "EXPT" in figure 2, and proves to be larger than the theore- tical 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, diffuse- ness of the droplet surface, which affects u(r), and momentum exchange arising from elastic scatte- ring 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 presen- ted an approximate model of the atom-droplet inter- action which predicts values of the mobility in reasonable agreement with experiment when the droplet size is calculated using macroscopic ther- modynamics. Further refinements are suggested which should improve that agreement.
References
/I/ Henson, B.L;, Phys. Rev. (1977) 1680. /2/ Edwards, D.O., Fatouros, P., Ihas, G.G.,
M~ozinski, P., Shen, S.Y., Gasparini, F.M.,and Tam, C.P., Phys. Rev. Lett.
