IONLoss BY DIFFUSION
265
Ion Loss by Diffusion in the Radiolysis of Gases1
by Cornelius E. Klots and Verner E. Anderson Health Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee
(Received M a y $0, 1966)
Expressions are presented for calculating the role of surface neutralization in gases homogeneously irradiated in spherical vessels. It is found that diffusion will be ambipolar in typical situations. Comparison with experiments suggests that uncontrolled effects, probably thermal convection currents, apparently enhance the role of surface neutralization beyond that calculated.
It has been recognized for some time that the site at which neutralization occurs can have a bearing on the chemical products from gas-phase radiolyses. The clearest indication of this is seen in the effect of weak externally applied electric fields on the various chemical yields.2* More recently dose-rate effects have been plausibly interpreted in terms of a similar alteration of the neutralization mechanism.2b Thus at low r a d h tion intensities it is suggested that charged particles diffuse to the vessel wall prior to neutralization; at much higher intensities bimolecular volume combination can occur. LET effects in gases may be accorded a similar interpretation in terms of effective dose ratesS3 It should be useful to have explicit formulas from which the fraction of ions terminating at each site can be calculated at a given set of experimental conditions. This would facilitate not only t,he evaluation of existing data but the design of experiments in which laboratory parameters are deliberately varied so as to encompass both modes of neutralization. Simple order-of-magnitude considerations suggest that the fraction of volume recombination, f, is given by f/(l - f ) 2
=
where the dimensionless parameter ~2
(1)
K2
K
r+ = r-
= R41a/D+D-
-VD+On+
= -VD_On-
+ Ep+n+ - Ep-n-
in which Die, p i , and ni are, respectively, ionic diffusion coefficients, mobilities, and concentrations. The first two are related at the low fields of present concern by DiO
is defined by
in which R is a characteristic dimension of the radiation vessel, I is the rate of ion-pair generation per unit volume, is the bimolecular recornbination coefficient, and D+ and D- are appropriate d8usion coefficients. In what follows below we shall see that K does indeed emerge as a convenient analytic parameter although eq 1 itself is neither unambiguous nor particularly accurate. (Y
The model to be considered is both tractable and quite general. It has been investigated, for example, in connection with certain photochemical4 and pyrolytic6 reactions. The gas is assumed to be under uniform irradiation in a spherical vessel of radius R. Since volume recombination is being considered the treatment is limited, it would appearj8 to pressures less than about 200 atm, above which parent-ion recapture sets in. Ions reaching the walls are assumed to be neutralized there without reflection. It is further assumed that the charged species of a given sign are of one type; thus, if the medium is capable of electron attachment, this is presumed to occur quite rapidly before the free electrons diffuse significantly. Finally, the treatment below will assume a steady state. One has for the fluxes
~
_
_
_
=
pilcT/e
_
(1) Research sponsored by the U. S. Atomic Energy Commission under contract with Union Carbide Corp. (2) (a) H. Essex, J . Phys. Chem., 58, 42 (1954); (b) T. W. Woodward and R. A. Back, Can. J . Chem., 41, 1463 (1963). (3) R.A. Back, T. W. Woodward, and K. A. McLauchlan, ibid., 40, 1380 (1962). (4) R. M. Noyes, J . Am. Chem. SOC.,73, 3039 (1951). (5) R. M. Marshall and C. P. Quinn, Trans. Faraday SOC.,61, 2671 (1965). (6) S. G . ElKomoss and J. L. Msgee, J . Chem. Phys., 36,256 (1962).
Volume 71, Number 8 January 1967
266
-
CORNELIUS E. KLOTSAND VERNERE. ANDERSON
as a function of the single parameter K. The relation obtained is displayed in Figure 1. The severe limitations of eq 1 will be evident. It remains now only to show how the “effective diffusion coefficients” Di can be obtained. From its definition D- is given a t the center (.$ = 0) by
.8
a W
5
/
I
.4t-
:I-
-
‘:I&/
D-/D-O = 1
e2 n-(n+ - n-) + __ T V 2n-
(4)
E&
With manipulation and use of the identity
1.0
loo
10.0
IO1
D-/D-Q
+ D+/D+O = 2
this rearranges to
K
Figure 1. The fraction of ion pairs terminating in the volume as a function of the parameter K (defined in the text).
D-/Da =
D-O D,
+ [2e2D+oD-o/eo~kT]F(~)(5) + [2e2D+oD-0/eoakT ] F ( K)
where F ( K ) = ( q / t ) 2 / [ ~ 2- ( q / ~ ) ~ ] ~ - ao ,function of K only. The form of eq 5 follows one of Allis and brings out the limiting values of D-. Evaluation of F ( K )from the machine solutions indicates
The space-charge electric field is given by V E = (e/eo)(n+ - n-)
Finally matter conservation gives
F ( K )‘V ~ ~ / 3 6
Our method of solution will be to assume that, for the ions of either sign, a well-defined diffusion coefficient existas,independent of position, defined by
ri =
-DiVni
This is clearly correct in the absence of a space charge when Di = DiO and is also correct in the ambipolar or high space-charge limit when
D+
==
+ D-’)
D- = D , = 2D+oD-o/(D+o
It follows, in intermediate situations, from the further assumption of proportionality discussed by Allis.? Utilizing the result D+n+ = D-n- and the substitutions
aRm+
-
aRm-
q = D - -D+I = r/R Equation form
may be transformed into dimensionless
(d2q/df2)
+ ~ ‘ -1 v2/E
= 0
(3)
with a convenient set of boundary conditions q(I =
0)
=
v(5
= 1) = 0
Equation 3 has been solved numerically with a CDC1604 computer to get the flux a t the wall and thus f, the fraction of ion pairs combining within the Volume The J o u T of ~ Physical ~ ~ Chemistry
This simple result, together with eq 4 and 5 , Permits the evaluation of the effective diffusion coefficients a t any value of K . Via a reiterative procedure, K and thus the fraction of volume recombination appropriate to the conditions may be deduced* The simple form of F ( K )in eq 6 may also be obtained from a variational treatment of eq 3 using as trial function a solution appropriate to small recombination.8 We find, in fact, that it holds quite well for K 5 25. This is entirely sufficient since insertion in eq 5 of parameters appropriate to pressures about 1 atm indicates that diffusion is ambipolar down to the lowest values of K of interest. Only under quite pathological conditions is there apt to be any ambiguity in the “effective diffusion coefficients’’ for K > 25. This result, that gas-phase radiolyses are almost always conducted under ambipolar conditions, may be examined in another light. Consideration of eq 5 shows that diffusion will be ambipolar so long as R > I where 1 is the Debye length. Since this condition holds quite typically, gas-phase radiation chemistry may legitimately be thought of as a sort of plasma chemistry. The boundary condition q ( ( = 1) = 0 is, as is well known, an oversimplification. At a plane boundary one has for the outward and inward fluxes, respectively (7) W. P. Allis, “Handbuch der Physik,” Vol. XXI, Springerverlae. Berlin,. T)_ 397 E. (8) R. H. Ritchie, private communication.
IONLoss BY DIFFUSION
267
Di-dni - 3n.D. - -
ri+
1
1
2 dr
4Xi
+ -Di2- dni dr
ri- = 3niDi 4Xi
where X i is a mean-free path. Then defining a surface accommodation or sticking coefficient by u = 1
one obtains ( 5 = 1)
fol.
-
ri-/ri+
a more correct boundary condition
Acknowledgment. The authors wish to acknowledge several useful discussions with Dr. R. H. Ritchie and Dr. H. C. Schweinler.
Appendix Ionic diffusion coefficients can apparently be calculated with sufficient accuracy from the simplified Langevin formula for ionic mobilities at standard temperature and pressure pi =
Thus will vanish within a few mean-free paths of the boundary unless a is very small. Use of eq 7 can only increase the extent of volume neutralization beyond that calculated above. The available experimental data hardly justify any such concern, as we shall now see. A comparison of the present results with the systematic investigation of Back, et u1.,2bis invited. We estimate, as outlined in the Appendix, D+O and D-0 in propane a t his densities to be 3 X and 2 X 102 cm2/sec, respectively. With cy cc/ion sec and R 3 cm, his data indicate surface neutralization at dose rates where, according to Figure 1, neutralization should have been almost entirely within the bulk volume. A simple reconciliation can be achieved only by assuming improbably low values of cy. Nevertheless the correctness of Back’s interpretation seems indisputable; his data are of the form of Figure 1 and are consistent with other jnvestigati~ns.~It is the lack of quantitative agreement with the present calculations which is disappointing. While there is some ambiguity in the geometry of Back’s irradiation vessel, another origin of the discrepancy is suggested by a consideration of some recent work of Bone, et aZ.1° Using dose rates much higher than even the highest of Back, et al., these authors present evidence that neutralization occurs predominately at the walls. This apparent incompatibility among the experimental results and the failure of the present model t o describe them can be most simply understood in t m n s of a role of convection currents, especially a t large radiation intensities. Similar effects seem to arise in photochemistry4 and are extremely difficult to encompass in any theoretical model. Temperature gradients will thus, we suggest, have to be
-
minimized if it is ever desirable to eliminate or control the extent of surface neutralization.
-
3 ~ . 9 ( c y ~ ) - ’ ”[cmz/v sec]
where a is the polarizability of a bulk molecule in units of uO3(where a0 is the Bohr radius) and M is the reduced mass, in atomic units, of the ion-neutral pair. Correction of this standard mobility to the appropriate experimental density and application of the NernstEinstein relation should then yield a quite acceptable estimate of D,O. Understandably no such formula is available for electron-transport parameters. Nevertheless examination of drift-velocity data indicates11j12electron diffusion coefficients are, with rare exceptions, in the range 1-4 X lo2 cmp/sec a t standard conditions. Thus uncertainty is quite acceptable since, as noted above, diffusion will usually be ambipolar, or nearly so, and thus insensitive to D-O. By far the greatest source of uncertainty in calculations of the present sort lies in estimating the secondorder neutralization coefficient. Little in the way of generalizations can be offered. Recent measurements of electron-molecular ion combinations indicate a 2X to 2 X cc/ion sec.13-15 KOpressure dependence of these coefficients has been established. The mechanism of ion-ion neutralization is somewhat more ~ o m p l i c a t e d ’but ~ * ~effective ~ second-order combination coefficients in the above range are again indicated.
-
(9) G. R. A. Johnson and J. M. U’arman, Trans. Faraday Soc., 61, 1709 (1965). (10) L. I. Bone, L. W. Sieck, and J. H. Futrell, J . Chem. Phys., 44, 3667 (1966). (11) G. S. Hurst, L. B. O’Kelly, E. B. Wagner, and J. A. Stockdale, ibid., 39, 1341 (1963). (12) T. L. Cottrell and I. C. Walker, Trans. Faraday Soc., 61, 1585 (1965). (13) H. J. Oskam and V. R. Mittelstadt, Phys. Rev., 132, 1445 (1963). (14) W. H. Kasner and M. A. Biondi, ibid., 137,A-317 (1965). (15) R. C. Gunton and T. M.Shaw, ibid., 140, 8-756 (1965). (16) B. H. Mahan and J. C. Person, J . C h a . Phys., 40, 392 (1964). (17) T. 5. Carlton and B. H. Mahan, ibid., 40, 3683 (1964).
Volume 71, Number 2
January 1967
