Transport properties of methane

Physica

46 (1968)

539-549

TRANSPORT

o North-Holland

Publishing

PROPERTIES

Co.,

Amsterdam

OF METHANE

G. A. STEVENS r FOM-Instituut

VOOYAtoom-

en Molecuulfysica,

Amsterdam, Nederland

Received 16 July 1969

synopsis Strongly simplified expressions for the transport cross sections, including inelastic collisions, are calculated numerically for methane izCH4 and the mixture rsCHq/rsCHq. It is found that viscosity is hardly influenced by inelastic effects. For diffusion the correction is found to be much less than experimentally observed, but of the correct sign. The thermal diffusion factor shows a large dependence on the value of Zrot and seems to be of the right order of magnitude. The calculated value of the ratio (Zrot)cD,/(Zrot)cn, at 300 K is 0.71, comparable with the experimental outcome of 0.6.

1. Introduction. Although kinetic for molecules with internal degrees Uhlenbeck-de Boer treatment of several years now L 2)) the numerical

expressions for the transport coefficients of freedom based on the Wang Changthe Boltzmann-equation, are available

values are still lacking. This, obviously, is due to the enormous increase in complexity of the calculations of the differential cross sections by the introduction of inelastic collisions. In a previous papera) the author has tried to simplify the scattering as far as necessary, to be able to undertake such calculations. The expressions were set up for molecules with a tetrahedron-shape, while only the coupling of the rotational angular momenta was taken into account. Although these expressions can easily be modified to be valid for other types of molecules, the computation has to be directed much more to specific cases. This stems from the fact that additional parameters must be introduced in comparison with the elastic scattering case. One of them determines the ratio of collision and rotation time, the others are the different moments of inertia involved. The calculations presented here were centered round methane and some of its isotopes. The results of these calculations, as several approximations are involved, of course only serve as a suggestion in unravelling the whole complex of transport phenomena for molecules with internal degrees of freedom. $ Present address : Laboratorium voor Ruimteonderzoek,

Nederland. 539

Rijksuniversiteit,

Utrecht,

540

G. A. STEVENS

2. The collision. integrals. Transport coefficients are expressed in terms of so-called collision integrals 1’ 233). These are weighted averages of some functions overideflection angles, kinetic and internal energies. For the case of methane, where a rotational level j is (2j + 1)2-fold degenerated, this average reads :
x)jqg, =

~

x

1)2 exp(--Epc - Eqj,) x

(2i + 1)2 (2j+

Do

1

exp(-g*“/T*)

x3-j

g*” dg*” x

0 x x

(1)

Fu,,tz(g*'/T*, 2) ct~+kt(g*', x) sin x dx. s 0

The various quantities q, q’ I%, E 5 g *’ c X

have the following meaning:

are the species to which the colliding molecules belong. is the reduced mass. is the reduced rotational energy of q in state i (epl = E,,/kT). is the reduced temperature (T* = kT/c). is the reduced relative kinetic energy (g*” = pqq,gs/2E). is the well-depth of the spherically averaged potential between molecules. is the deflection angle.

Qq and Qqf are normalization constants the internal states, defined by:

for the probability

the

of occurrence

of

Qq = 2 (2i + 1)2 exp(--Ep,). i u2,_,kl(g*a, x) is the degeneracy-averaged process ij -+ kl. The approximation

C k,l

differential

used in the calculations

s s

sinx dxFi3,kt(g*“/T*,x) oij+kl(g*",

cross section

for the

consists of the assumption

:

x) m

0

co

co

b dW&*“/T*, x(b, g*“))+

m

s

b dbPij_,k@*‘, b) x

0

0

x CFij,k@**/T*,

&-

x(b, gka))-

F,,(g*“/T*,x(6’,g*"))l,

(2)

TRANSPORT

PROPERTIES

OF METHANE

541

where :

F,,__denotes F~J,6~. ~(b’, g*“)

is the deflection

x(b, g*%)

meter b’ = bJ[*“/g*’ and reduced kinetic energy g*a. is the deflection angle for an elastic collision with impact parameter b and a “mean” reduced kinetic energy g*’ defined by:

g*n =

pi& kl

g*2 -

angle for an elastic

(Eqk + E,,, -

;

is the transition

probability

E,, -

collision

with impact

E,,,) = g*” -

para-

-$

of the process ij --f kl (AE # 0).

As the transition probabilities were calculated in first order, F,, was independent of i and j. The spherically averaged potential was assumed to be a Lennard- Jones (12, 6) type of the form : VI)(Y) = 4E

[(G),,

-

(:)“I.

The “perturbing” potential, as far as the r-dependence is concerned, was taken to be equal to c?~E(G/Y) 12, the angle-dependent factor being a linear combination of third order Wigner-functions as given in ref. 4, denoted by E3k&4-

The transition

where : ro is the distance

probabilities

were approximated

in the following way:

of closest approach.

a=

This expression originates from a time-dependent perturbation approach, where the zero-order orbit with kinetic energy t*2 and impact parameter b,

l

is replaced by a straight one through 70, with uniform velocity g = &@$. For the full details of the approximation mentioned the reader is referred to a previous paper 4). Writing y2 = g*“/T*, the most important functions, Fi~,kl, which occur are :

542

G. A. STEVENS

, which together with the viscosity

determines

Zrot, roughly the

ratio of elastic and inelastic collisions.

W y2-

&‘(Y~-$)

cos x, which determines diffusion,

-

toss x, which together with a) determines viscosity.

c)

Y4-Y2(P

$)

4

yz(y2~~‘(yZ--~)cwi) and (%,-w(,s-

Jy+s-$+sX)>

which determine

the temperature

behaviour

of the diffusion factor.

3. Methods of calculation. As P+.kl is a function of g*’ and not of g*‘, the integral over g*’ was transformed to an integral over g*a. The weight functions being exp(--g*“/T*) the integration was carried out by a GaussLaguerre quadrature. The abscissae chosen were those of a 32-points integration with weight function exp(-e*“) 5). The integrant being equal to J, the integral was worked out according to: 0r J exp( -g*“/T*)

dg*” = L J’ exp( -gka) dg*%,

where J’ is defined by: _

= 0

k&_ + (’ -Fc’ ‘**)

if

g*’ >

otherwise.

I_!$ I,

(4)

In this way it was possible to carry out the integrations over g** for T*E[ 1,5] with one set of 32 abscissae and weights. The integrants themselves are integrals over the impact parameter. As is done in ref. 6 the integration over b was transformed to one over z, which is a function of r$ = Q/CTand g*8 defined by:

The value of z varies between 0 (b = co) and 1 (b = 0). The advantage of this transformation is that one is not forced to calculate rt for each pair of values for b and g-*‘. A disadvantage is that for g*a < 0.8

TRANSPORT

a discontinuity

PROPERTIES

OF METHANE

543

appears in the values of z, due to orbiting. The integral then

has to be split up into two parts, one from 0 to ze and another from zr to 16). The integration method chosen was that of Gauss-Legendre. The number of abscissae varied from 32 for high values of g*’ and AE, to 80 for low values of g*’ and AE. The transition probabilities P+kl were calculated with eq. (3). The parameter a is expressed in g*’ and z according to:

It is a rather lengthy procedure to put in from the beginning of A occurring, as too many of them are involved. Moreover, would be only useful for one specific kind of molecule. So a set values of A was selected, ranging from 0 to 200, for three uJG/&

= 17 (25, 2512, 25/3), between

The interpolation

all values the result of twenty values of

which later should be interpolated.

formulas used were:

as to

A:

exp(al

as to

11:

a5(rY”.

In A + azA* + asA& In A + adA+), (6)

The deflection angles x were calculated by a method described in ref. 7, where a Gauss-Mehler quadrature formula is recommended. For the angle ~(6, g*‘) this can be performed without any trouble as its z- and g*“-values are the same as those occurring in P+M. The other angle ~(b’, g*“) requires a preceding determination of zj belonging to each abscissa z3 of the integration over z. This was done by determining r$ for an elastic collision with the same angular momentum, but with reduced kinetic energy g*’ = = g*’ + AE/2c. While P +kl only depends on the absolute value of AE/E, the deflection angle ~(b’, g*“) also depends on its sign. So the calculation had to be carried out twice. 4. Results. As a first step expression (3) for the transition probability was tested by the calculation of the ratio of Zrot for CD4 and CH4 at T = = 300 K. The values of the parameters used were: E/k=

150K,

(r = 4.22 A,

IcH, = 3.2176 x lo-40 g ems.

Icn, = 6.4302 x 1O-40 g cm2,

G. A. STEVENS

544

Fig. 1. The inelastic contribution C to
This ratio is independent of the asymmetry parameter 6 and does not involve any deflection angle. The calculated value of (Zrot)cn,/(Zrot)c~, was 0.7 1, which is not too bad in comparison with the 0.6 outcome from measurements on acoustical phenomenas). Next, the difference with the “elastic” values of the transport cross sections were computed as far as the integration over impact parameter and reduced kinetic energy was concerned. A typical example of the results is given in fig. 1, where this difference for (ys - (y, y’)> is plotted as a function of A for TX = 2 and q = 2512. As a test the temperature derivative of the inelastic contributions to was used. On the one hand this can be estimated from a plot of these contributions KS. T*, for one value of A, on the other hand it can be calculated from the relation: 2 dA
= _5A

2* ~7)).

~2

_

(y,

y’))

+

(7)

It turned out that above an energy change of about 0.1~ the relation was satisfied. Even an increase of the number of abscissae in the integration over z till 80 and in the Gauss-Mehler quadrature till 100 did not essentially improve the situation below 0.1 E. It was decided therefore to use these results only for the case of collisions between rsCH4 and isCH4 or isCH4. The minimum energy transfer in this case amounts about 0.16~.

TRANSPORT

After summation

PROPERTIES

545

OF METHANE

over the internal states of this molecule

of the results was checked

again by an equation

analogous

the reliability to (7), which

reads : 2

d e2 - 0,~‘))

= --s
dlnT* +

2

<(El,

+

E2, -

Cl

-

“2)(y2

(79 Y’)> + 2
(y,

(Y, 7’))) +

7’))).

(8)

Eqs. (7) and (8) were chosen because they are crucial in a judgement concerning the results for thermal diffusion. Both eqs. (7) and (8) being satisfied the last step was the interpolation between the values of 7, for different values of (&)cn, at T* = 2, which uniquely determines the asymmetry parameter 6 as shown in table I. TABLE I Various quantities derived from the calculations for several values of (Zmt) CH,

6

Z rot

6

5 10 12 15 20 00

0.83 0.59 0.53 0.48 0.41 0

(E/k)diff

137 144 145 146 147 150

Gl,.

1.59 1.26 1.21 1.16 1.11 0.93

(+)th.diff.

256 203 195 187 178 150

is the asymmetry parameter of the angle-dependent part of the potential: 0 1s Ir,(,; C$y) = 6 4E 7 E3(+). ( > is the potential-well-depth derived from the self-diffusion coefficient, if the results are interpreted in terms of elastic collisions with a Lennard- Jones (12, 6) potential.

(Emdire.

is the inversion temperature of the thermal diffusion factor, where E is Glv. = kT/c the well-depth of the spherically averaged potential. is the apparent potential-well-depth, if the inversion temperature T&,. is put equal to 0.93, valid for the elastic scattering case with a Lennard-Jones (12, 6) potential.

(c/k)th.diff.

The results will now be discussed in connection with the transport properties diffusion, viscosity and thermal diffusion. The asterisk denotes a reduction of the quantities concerned, with the corresponding values in the elastic scattering case of rigid elastic spheres with diameter u. 4.1. Diffusion. The calculated values of the inelastic contributions to * are plotted in fig. 2, for some values of Zrot. All values are

G. A. STEVENS

546

negative and tend to go to zero as T* increases. So a plot of
of this effect,

the quantity

d ln * dlnT*

is plotted in fig. 3 as a function of T* for Zrot = 5 and 00. This picture predicts for measurements of the diffusion coefficient of isCH4 between 300 K (T” = 2) and 375 K (T* = 2.5) an e/k value of 135. These apparent values are also given in table I. The effect is rather small in comparison with the experimental data, which gave a preferred value of 70 K 9). 4.2.

Viscosity.

(
In fig. 4 are shown the inelastic

(y, Y’)~> -

contributions

to

Q )*

as a function of T*. Because of the compensation of the two terms, these values are smaller than those for *. Moreover, above T* = 2 the contributions are almost constant and as a consequence do not affect the temperature behaviour of the viscosity, a fact which seems to be affirmed by experiment.

0

I 1

1

I

I

3

4

-T*

Fig. 2. The inelastic contribution C to (~2 - (y, y’))* for the mixture 12CH#sCH4 as a function of T* for Zrot = 5, 10 and 15.

TRANSPORT

PROPERTIES

OF METHANE

Fig. 3. The logarithmic derivative of (72 - (y, 7’))” and 00. D =

as a function of T* for Zrot

2d ln
11005

0

- 0.002

1

,

3

Fig. 4. The inelastic contribution C to ( T* for Zrot = 5, 10 and 15.

, 4

-T*

6 <&>2)* as a function of

G. A. STEVENS

548 a M2

(101

0

- 0.0 0 5 1

Fig. 5. The thermal

diffusion

I

I

2

I

3

4

factor

OLfor the mixture

lsCH#sCH4

T* for Zrot = 5, 19 and co, in the limit of zero concentration

-T’

as a function

of

of 13CH4.

4.3. Thermal diffusion. The formula used for the computation of the thermal diffusion factor is the one given in ref. 10, formula (8), which was linearised in the relative mass difference and the inelastic contributions. The result is shown in fig. 5. It shows a large dependence of the inversion temperature upon the value of Zrot, which is mainly due to the inelastic contributions in C” =
(y, y’))>/3
(7, 7’)).

These inversion temperatures are given in table I, together with the c/kcase. An experivalues, connected with T&. = 0.93 as in the “elastic” mental outcome for (E/k)th.diff.form experiment is 194 K, which corresponds to Zrot = 12, which is exactly the value reported in ref. 8. 5. ConcZusio~zs. Although table I shows that the asymmetry parameter 6 is not so small that one might speak of a perturbation character of the angle-dependent part of the potential, the scattering model used may work. In all cases the inelastic corrections seem to have the right sign and, except for diffusion, the correct order of magnitude. The calculations themselves are rather lengthy, because of the accuracy required. Moreover, the integration variable z seems not to be an adequate one, especially for small energy transfers. Finally the calculations presented here may be compared with the results of Sandler, Dahler and Mason11p12y l3), concerning the transport

TRANSPORT

PROPERTIES

OF

METHANE

549

properties

of loaded spheres. Obviously methane cannot be regarded as a sphere. Moreover, the model is not very suited to describe the

loaded temperature behaviour of transport properties. Nevertheless, one may expect that the dependence of the transport properties in the Pidduck apAs to proximation, on the value of Zrot should be the same, qualitatively.

the diffusion (Table I in ref. 12) there is agreement in the sign of the corrections and their increase with decreasing Zrot. Moreover, their calculations show a much smaller correction on viscosity (Table III in ref. 1 l), than on diffusion which agrees with our results. A last remark should be made upon thermal diffusion. From the calculations presented in refs. 12 and 13 one might conclude that the main inelastic contribution comes from the term Q$“’ or Se, - 5 (ref. 10). The calculations presented here, however, indicate that for mixtures with different mass, the main effect comes from the corrections on 6C$ - 5, which even are compensated to some extent by the terms containing the “pure inelastic” factor 6Cij - 5. The author would like to thank Mr. F. Vitalis Acknowledgements. for his help and advises in the computation problems. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation for Fundamental Research on Matter) and was made possible by financial support from the Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek (Netherlands Organization for the Advancement of Pure Research).

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