A simple momentum-transfer theory for the composition dependence of ion mobilities and diffusion coefficients in gas mixtures at arbitrary field strengths is corrected, extended, and compared with a similar theory based on momentum and energy transfer, and with results based on direct solution of the Boltzmann equation by Kihara's method. Final equations are recommended for predicting composition dependences, given only results on ion mobilities and diffusion coefficients in the pure component gases

Mason, E. A.

Robson, R. E.

Whealton, J. H.

English

NASA Technical Reports Server

COMPOSITION DEPENDENCE OF ION TRANSPORT
COEFFICIENTS IN GAS MIXTURES
"J. H. Whealton and E. A. Mason H
Brown University, Providence, Rhode Island 02912 Hm
= 0
R. E. Robsont tZ
Institute of Theoretical Physics, University of Alberta bw
Edmonton 7, Canada n Wo
o K)
g ON
* dD
ABSTRACT H 0
&H
Simple momentum-transfer theory for the composition dependence I Z o
of ion mobilities and diffusion coefficients in gas mixtures at He
arbitrary field strengths is corrected and extended, and compared
with a similar theory based on momentum and energy transfer, and ~ t
with results based on direct solution of the Boltzmann equation by
C
Kihara's method. Final equations are recommended for predicting
composition dependences, given only results on ion mobilities and
diffusion coefficients in the pure component gases.
Ln :3
C3C7P C-J P
https://ntrs.nasa.gov/search.jsp?R=19740004277 2020-03-23T12:35:47+00:00Z
-2-
Simple momentum-transfer theory has recently been used to pre-
dict the composition dependence of ion mobilities1 and diffusion co-
efficients2 in gas mixtures at arbitrary field strengths. The pur-
pose of this comment is threefold: to extend the range of validity
of the results of Ref. 2 to higher fields by allowing the ion tem-
perature to be anisotropic and to differ from the gas temperature;
to correct an error in the expression used for the ion energy that
affects the final results of Refs. 1 and 2; and to point out the
relation of these results to a similar theory developed by Milloy
and Robson,3 and to recent work based on direct solution of the
Boltzmann equation.
By equating the momentum transferred to ions of charge e by an
electric field, E , to the momentum transferred by the ions to the
neutrals through collisions,l and by taking into account the pressure
forces that arise if the ions are not distributed uniformly in
space,2 one obtains the following momentum-balance equation for the
mean ion velocity () ,
>' 4*
En v) iP.N. (v .Q(v )> = ne - pi (1)
where C is a proportionality constant of order unity, n the
number density of ions, i. the reduced mass of an ion-neutral
pair, N. the number density of neutral species j , v . the
relative speed of a colliding ion-neutral pair, Q(v .) their
momentum-transfer or diffusion cross section, and pi the partial
pressure tensor for the ions. Pointed brackets indicate an average
over relative speeds. In a coordinate system in which E is directed
-3-
along the z-axis, p.i is a diagonal tensor with components
Pxx = pyy = _L and z = pl . Assuming that the ions obey the
ideal gas equation of state, we introduce
pll = nkT , pl = nkT , (2)
where the temperatures parallel and perpendicular to the field are
defined by
kTII = m [(vz2>  (z 2  , kTL = m (vx2>) , (3)
where m is the ion mass. Then V = k .
We focus attention first on the mobility by considering spa-
tially uniform conditions, for which ( = vd E KE , where
vd is the drift velocity and K the mobility. From Eq. (1)
we can write K in terms of the mobilities K. in the pure com-
ponents,
x. (v .Q(v .))1 i( r r > (4)
K K. (v .Q(v .)>.
where the x. are mole fractions, and ( )> means an average in
pure gas j , as distinguished from an average in the mixture ( ).
The evaluation-of the ratios (vrjQj )/(vrjQj> j proceeds as in
Ref. 2, and involves three approximations. First, the average of
the product, (v rQ(v .)) , is decomposed into the product of the
averages, (vrj) Q((v< rj) ). Second, the average (vrj) is re-
placed by the rms average (v r2 1/2, which is then evaluated
from the partitioning of the ion energy among thermal energy, drift
energy, and random field energy. The third approximation enters
through the expression for the energy partitioning,
(vrj2)>  (v2  t (v 2 ) = 3kT/. + <(v 2  [1 + (M)av/m ()A ) Z
-4-
where V. is the neutral species velocity, M. its mass, T
3 j
is the gas temperature, and (M >av is a mean mass of the gas
mixture, defined as
(M) EW.M./E. , (6)
>Mav j j j 
. x.M. (v ( ))/(m M) (7)) j j r j
The energy partitioning can be calculated by Wannier's method,
as was done in Ref. 1, or equivalently by an energy-balance
equation, as was done in Ref. 3. Unfortunately, there is an error
in the expression for the energy partitioning in Ref. 1--in Eqs.
(27) and (29) the Qj should be replaced by (vrjQj ) to give
the correct weight factors w. as in Eq. (7). The numerical
example in Ref. 1 is therefore incorrect in detail because of the
incorrect w. used. However, at high fields the relative speed
is the same as the ion speed, and there is then no mistake in the
The theory of Milloy and Robson3 for the mobility is essentially
the same as the foregoing treatment up to this point, but makes
further approximations in order to express the deviations from Blanc's
law entirely in terms of the behavior of the ions in the pure neutral
components, without explicit reference to the mixture itself. Thus
the mixture averages are expanded in series
Krj <Vj2 d<vvdj ( j > )>j
2 d<vr .j2 3 + * (8)2 r 2
vd> vj j d > j(
-5-
From Eq. (5) we find
(vrj 2 > - (v j2j Vd2 [1 + (M)v/m] - Vdj 2 (1 + Mj/m) . (9)
Since this expression appears as a correction term in Eq. (8), it
need be evaluated only in lowest order, corresponding to
rvj Kvrj Q j , which yields
1 -1
MC'\r .> <vj2> D - -]
vdj Vdj(m + Mj.
2
- (m + Mj) Vdj . (10)) dj (10)
The derivative in Eq. ( 8) is obtained by differentiation of Eq. (1)
written for a single pure component having the same total number
density as the mixture (Nj = N), and is found to be
din <v Q(v) . _d n vd 1 i ~ dn v
d v > d 2  M+MM d In (E/N) d In (E/N)S- 
--
d In K.2v
n K . (11)
2 m + M d in (E/N)
2 Vdj 3
Substituting these results back into Eq. (4), we obtain
x. x. d In K.
. - 1; 1 -1 1* ( 1 - A.) (12)
j j j d In (E/N)
A- 1 = (m + M) K 2 (m + .)K.) _ (13)i (m + K
-6-
which is the result of Milloy and Robson. This expression is re-
markably similar to one obtained from a direct solution of the
Boltzmann equation by Kihara's method. The main difference.is that
the tAj in Eq. (12) is multiplied by a complicated factor G. that
happens to be numerically close to unity. Thus Eq. (12) can be recom-
mended as an excellent estimate for deviations of the mobility from
Blanc's law.
The preceding calculations can obviously also be applied to the
longitudinal (DIi) and transverse (DL) diffusion coefficients,
but an additional feature must enter. It is clear from Eq. (1)
that spatial inhomogeneity affects < ) . Since Eq. (5) shows
that (vrj) is related to the mean ion speed, spatial inhomo-
geneity must also affect (v r. . Thus we must allow both
(- ) and ( vrj) to have small contributions from spatial inhomo-
geneity, but we may consider these as perturbations and linearize
with respect to them. Denoting the spatially homogeneous case
with a superscript, we expand the average (v .Q(v .)
in a series similar to Eq. (8),
(v.rQ(vrj) = (vrQ(v )) +
+r 2 d (v .Q(v r) )o
2 - (v )i d v o + ... . (14)
From Eq. (5) we find, on setting (v z ) = d 
(rj2 -. 2 ( 2 vd [1 + (M )a/m]
- 2vd  1 av vd.kT nm neE 9z (5
-7-
the last step following by substitution for (vz) - Vd from Eq.
(1). Since this last result is independent of j , we can multiply
Eq. (14) by vjNj and sum over j.; the derivative term is then
obtained by differentiation of Eq. (1), yielding a result similar
to Eq. (11),
d In El.N. (v .Q(v J) >1
r i1 m d In K 1S2 m+ M d In(E/N)
d (v rj 2vd2 + av
Substituting these results back into Eq. (1) and linearizing with
respect to derivatives of n , we identify the transport coefficients
and observe the following relations among them:
eDI % kTi + d In K (17)
K d In(E/N) "
E n kT( . (18)
Equations (17) and (18) have been derived previously by pheno-
menological arguments,5'6 and have been shown to give good quali-
tative agreement with numerical solutions of the Boltzmann equa-
tion.6 A more exact treatment according to kinetic theory gives
eD
=D , l k T d In K + " (19)K 
- k T,1 d In (E/N) ,
where y = 3/2 and L < 1/2 , which is valid to order (E/N) .
Within this order, the ratio DI /D_ is the same whether
y1I = 3/2 or 1 and 7Y = 1/2 or 0 are used. Thus Eqs. (17) -
(18) are consistent with Eq. (19), even though the individual
y values differ.
The ion temperatures are obtained from Eq. (3) by an energy-
partitioning calculation, either by Wannier's method applied to
-8-
mixtures,1 or from a first-order solution to the Boltzmann equa-
tion.7 In either case, the result for the mixture is
k5Tm + j VMd~a 2
kTI =kT + <Vd (21)
5m + 3 MA a v
where (M>av is given by Eq. (6) and
(MA av w M A. /Z . (22)
in which A is a ratio of cross sections of order unity.
To find the composition dependence of the diffusion coefficients
we substitute Eq. (19), which applies to both mixtures and pure
gases, back into Eq. (4),
TI : x ( (vr Q(v r) d InK
n d I K
x + y d ln(E/N) (23)
The important new feature is the appearance of the ion temperatures in
Eq. (23). The temperatures in the mixture and in the pure components
are not the same at the same value of F/N because the partitioning
of the ion energy among drift and random field components depends on
the nature of the ion-neutral collisions. One immediate consequence
of this is the recognition that the diffusion coefficients will deviate
from Blanc's law at high fields even for the Maxwell model of constant
-9-
collision frequency, for which the ratios (vr j )/( vrjQj )
are all unity and all derivatives of mobility with respect to E/N
are zero. Numerical calculations of deviations from Blanc's law
for the Maxwell model have been performed and the results are shown
in Fig. 1.
The ratios (vr Qj 0/ vrj Q )o in Eq. (23) can obviously
be evaluated by the same methods used for the mobility. Substitut-
ing these results into Eq. (23) and linearizing with respect to the
derivatives of mobility, in order to be consistent with the trunca-
tions in Eqs. (8) and (14), we obtain an expression that still con-
tains a mixture quantity on the right-hand side, namely the deriva-
tive d InK/d In(E/N) from the last factor of Eq. (23). In order
to evaluate this quantity in terms of pure component properties, we
differentiate Eq. (12), which leaves us with second derivatives of
K. with respect to E/N. These second derivatives can be approxi-
mated by appeal to the following symmetry argument. Since the gas
is isotropic K must be an even function of E/N; expansion in series
must therefore involve only even powers of E/N,
K = K(O) [1 + a1 (E/N) 2 + a2 (E/N) + ... . (24)
Differentiating twice and comparing results, we find
d2 In K d In K 42 /N) + terms of order (E/N) . (25)
d (ln E/N)2  d ln(E/N)
This expression holds for both mixtures and pure gases. Substitu-
tion of this expression then yields the final result for the com-
position dependence of the diffusion coefficients,
- 10 -
D1I j (D'L) J (DII)J ) r TL
x. d In K.
+ e Z _ 3 (1 - A) (1 + (26)2kTj j K d In (E/N) 2 (ll2
The ion temperatures in this expression are to be evaluated from
Eqs. (20) - (21), but with the average masses calculated with weight
factors w. approximated by making expansions as in Eq. (8),
x. x. d In K.
x. i+ 1 1d 1i (E/N)1- . (27)(+ M.) K. 2 7 7 d . n (EIN)
These results are nearly the same as those obtained by solution of
the Boltzmann equation. The differences are relatively minor: the
A. is missing the G. factors as in the mobility expression of
Eq. (22), the values of yll and yL are somewhat different, and
the Boltzmann expression does not include the small derivative term
in the expression for the m..
As an excellent estimate for deviations of the diffusion coef-
ficients from Blanc's law we recommend the following procedure: use
Eq. (26) with A. given by Eq. (13), with yll and YL given their
accurate kinetic-theory values, and with the w. given by just the
leading term of Eq. (27).
As a numerical example we consider K+ ions in an equimolar
mixture of He + Ar at 3000K, using as input data the measurements
on K+ in He3 and K+ in Ar.9  Deviations from Blanc's law for K
and for DII are shown as a function of field strength in Fig. 2,
as calculated from the Boltzmann equation and from Eqs. (12) and
(26). The shapes and magnitudes of these curves have been discussed
elsewhere 4; the important point here is the closeness of the
agreement for the two calculation methods.
In summary, the various theories for the composition dependence
of ion mobilities and diffusion coefficients in gas mixtures have been
shown to be consistent. Equations (12) and (26) (modified as
suggested above) are recommended for predicting deviations from
Blanc's law as a function of field strength.
- 12 -
REFERENCES
Supported in part by the National Aeronautics and Space
Administration under Grant No. NGL-40-002-059, and in part by
the National Research Council of Canada.
tPresent address: Joint Institute for Laboratory Astrophysics,
University of Colorado, Boulder, Colorado 80302.
Present address: Department of Physics, James Cook Uni-
versity, Townsville, Australia.
1 E. A. Mason and H. Hahn, Phys. Rev. A5, 438 (1972).
2 J. H. Whealton and E. A. Mason, Phys. Rev. A6, 1939 (1972).
H. B. Milloy and R. E. Robson, J. Phys. B6, 1139 (1973).
J. H. Whealton and E. A. Mason, Ann. Phys. (N. Y.) (to be
published).
C. E. Klots and D. R. Nelson, Bull. Am. Phys. Soc. 15, 424
(1970).
6 R. E. Robson, Australian J. Phys. 25, 685 (1972).
H. Hahn and E. A. Mason, Phys. Rev. A7, 1407 (1973).
8 E. W. McDaniel and E. A. Mason, The Mobility and Diffusion of
Ions in Gases (Wiley-Interscience, New York, 1973), Sec. 5-1.
D. R. James, E. Graham IV, G. M. Thomson, I. R. Gatland, and
E. W. McDaniel, J. Chem. Phys. 58, 3652 (1973).
/3
FIGURE CAPTIONS
FIG. 1. Ratio of D j(Blanc) to Dl1(mix) at high fields for
an equimolar Maxwell-model binary gas mixture, as contours in a
KI/K2 vs. M /M2 plane. We have taken m = M2 . Notice that
the deviations from Blanc's law are always negative. The corre-
sponding ratio for mixture mobility is always unity for the Maxwell
model.
FIG. 2. Percentage deviations from Blanc's law for mobility,
100 K-1 - K-1 K-1 and for D as a function of fieldBlanc / Blanc
strength, as calculated for K+ ions in an equimolar He + Ar
mixture at 300 0K. The solid curves are the results from the Boltz-
mann equation according to Ref. 4, and the dashed curves are the
present momentum-transfer results of Eqs. (12) and (26).
cDJ 0.25
30-
0.3
K
0.4
0. 6 0
0.9
.01 .03 .1 .3 1.0 3 10 30 100
M,/ M2
+20
0
-20
-40 -D
-60 -
-80-
10 100 1000
E/N (Td)


Composition dependence of ion transport coefficients in gas mixtures

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