3874
GEORGEA. MILLERAND RALPHL. BUICE,JR.
On the Knudsen Limiting Law of Thermal Transpiration
by George A. Miller and Ralph L. Buice, Jr. School of Chemistry, Georgia Institute of Technology, Atlanta, Georgia
(Receiaed June 14, 1966)
Transmission probabilities for gaseous free molecular flow under a temperature gradient have been calculated for various capillaries with a scattering law in which the faster moving molecule has a higher probability of being specularly reflected. This model can explain recent experimental observations of deviations from the Knudsen limiting law. Also the model indicates that the limiting law may be obtained approximately from isothermal free molecular flow data a t the two temperatures of interest.
I n working a t low pressures one must take into account often the thermal molecular pressure difference (TPD) which arises from thermal transpiration along temperature gradients in the apparatus. Thus in measuring the vapor pressure of a substance at low temperatures one must apply a correction to the reading of the manometer, which is normally at room temperature. There are two theoretical treatments of TPD which start from first principles, the Weber equation’ and the Dusty Gas modeL2 The latter, especially interesting in its bearing on rotational relaxation, is concerned with porous media. If extended to include capillaries, it is in essential agreement with the former. The Weber equation has, in its crudest form, no adjustable constants, since it is derived by joining Maxwell’s kinetic theory solution in the slip flow region to the Knudsen limiting law in differential form. It is, however, an approximate equation, and a consideration of data and certain details of the theory led Weber to adjust the constants and form somewhat. One of us3 has made a study of numerous T P D data in light of the Weber equation. It appears that the prediction of T P D is rather uncertain in the middle or transition region of pressure and that it is best to work at either higher pressures, where the effect is small, or at pressures sufficiently low that the Knudsen limiting law is approached to within a few per cent. However, whereas certain sets of data studied approached the limiting law within about 3% while following the Weber equation closely, other data deviated widely from the equation a t the same low pressures and left doubt as to the applicability of the Knudsen limiting law. More recently, careful experiments carried to very low pressures by Hobson The Journal of Physical Chemistry
and c o - ~ o r k e r shave ~ ~ ~shown a clear deviation from the law for the light gases helium and neon. I n this paper we present a theoretical model to explain these results, and we examine the consequences of our calculations in broader terms. The limiting law is directly traceable to the problem of free molecular flow. Clausing6 first recognized free molecular flow as a probability problem in which the collision density of gas molecules a t the solid surface is the important function. Referring to Figure 1, the rate of flow of molecules from reservoir 2 to reservoir 1 is equal to the rate at which they enter the capillary times the probability that they will leave through the exit. The latter quantity is commonly called the transmission probability (Q) and is a function of the dimensions of the capillary and the scattering law for the gas molecules a t the capillary wall. I n the free molecular flow region, then, TPD involves a steady state between two independent, opposing flows, and the limiting law is, in more general form
The subscripts to the
Q’S
refer to the direction of flow.
(1) S. Weber, Commun. Phys. Lab. Univ. Leiden, No. 246b (1937). (2) E. A. Mason, R. B. Evans, 111, and G. M. Watson, J . Chem. Phys., 38, 1808 (1963). (3) G.A. Miller, J . Phys. Chem., 67, 1359 (1963). (4) J. P. Hobson, T. Edmonds, and R. Verreault, Can. J . Phys., 41, 983 (1963). ( 5 ) T. Edmonds and J. P. Hobson, J . Vacuum Sci. Tech., 2, 182 (1965). (6) P.Clausing, Ann. Phys., 12, 961 (1932).
KNUDSENLIMITINGLAWOF THERMAL TRANSPIRATION
RESERVOIR 2
s
RESERVOIR I
m
h
r: 1;
0 L Figure 1. Diagram for the discussion of thermal transpiration through a capillary.
It should be noted that eq 1 contains the assumption that the number of molecules entering the capillary per unit time and area of opening from either reservoir is given by the usual kinetic theory expression, P/ (2nmlcT)'/'. Since we are dealing with a nonequilibrium situation in which at least a very small deviation from a Maxwell distribution must take place, this cannot be exact. We have put this difficult question aside in order to be able to study the more obvious aspects of TPD. Accepting t,herefore a Maxwellian distribution of velocities in either reservoir, we can say that the Knudsen limiting law is based on the assumption that Q12 = Q21, which will be true for certain simple reflection laws: (1) totally specular reflection, ( 2 ) totally diffuse reflection, i.e., the cosine law, or (3) a fixed degree of specular reflection, in which the impinging molecule has a fixed probability of being specularly reflected regardless of its previous history. In order to explain the experimentally demonstrated deviations from the Knudsen limiting law cited above it is necessary to show that QIZ
< Qz, where Tz > TI
(2)
which requires that the scattering law varies in some way with temperature. Unfortunately not enough is known about the details of the scattering of molecules at ordinary surfaces. Molecular beam experiments'-$ indicate that at this stage Maxwell's picture of a mixture of diffuse and specular reflection is still useful. On the other hand, flow experiments have brought out the importance of preferential back scattering, which is rather the opposite of specular reflection. The model we present below is therefore a compromise between simplicity and experimental fact as we know it. The Model. It is reasonably clear from the work of DeMarcus'O that a scattering law capable of explaining the inequality (2) will not be amenable to an analytical solution for Q. We have accordingly set up the problem as a Monte Carlo or random number calculation. Since the problem of the scattering law is
3875
far from settled, we thought it to be as important to gain some insight into the details of free molecular flow as to arrive at numerical values of &. Therefore we have split the calculation into two parts, endeavoring to solve as much of the problem as is possible analytically and performing the remainder by the Monte Carlo technique. Some accuracy is actually sacrificed by this procedure. For an example of a purely random number approach to free molecular flow, we refer to the calculations of Davis" on variously shaped ducts in which he assumed the diffuse law of scattering. Our model consists of the following points: (1) Gas molecules are either diffusely reflected with complete accommodation with the wall or specularly reflected with no accommodation. (2) The probability that a molecule of translational energy, e, be reflected specularly is given by
P(e) = 1 -
e-f/co
where eo is a sort of critical energy for specular reflection. (3) The distribution of energies of molecules impinging on the capillary wall directly from the reservoir, as well as of the molecules which have just undergone accommodation and diffuse reflection, is assumed to be of the simplified form
where e is the translational energy measured in units of IC, the Boltamann constant. This equation is easier to handle in the already time consuming Monte Carlo calculations and gives the correct qualitative behavior as long as the fraction of molecules for which E > EO is not large and as long as eo is looked upon as an adjustable parameter not to be predicted with any accuracy from first principles. The correct expression, however, would be that of an effusing gas -ee
-c/T
T2
Now imagine a gas flowing from a reservoir at temperature Ti into a capillary of the dimensions shown in Figure 1. The gas is sufficiently rarefied that there are no collisions in the gas phase. Following DeMarcuslo we define the following probability functions, ~
(7) F. C. Hurlbut, J. AppZ. Phys., 28, 844 (1957). (Si 9. Datz, G . E. Moore, and E. H. Taylor, Proc. Intern. Symp. Rarefied Gas Dyn., 3rd (196%),1, 347 (1963). (9) J. N. Smith, Jr., J. Chem. Phys., 40, 2520 (1964). (10) W. C. DeMarcus, PTOC. Intern. Symp. Rarefied Gas Dyn., 9nd, Berkeley, 1960, 161 (1961). (11) D . H. Davis, J. AppE. Phycr., 31, 1169 (1960).
Volume 70,Number 19 December 1966
3876
GEORGEA. MILLERAND RALPHL. BUICE,JR.
normalized to one molecule entering per second: s(z) is the diffuse collision density per unit length of capiIlary ; SI(Z), the density of first diffuse collisions by entering molecules per unit length; Sl(z),Lcsl(t)dt, the density of first diffuse collisions anywhere beyond x; r ( y , z), the probability that a molecule which has diffusely collided a t y will suffer its next diffuse collision at z; and f(x, L ) , the probability that a molecule which has diffusely collided at z will eventually leave through the exit rather than the entrance. Clausing's equations then take the form = s1b)
S(2)
Q
=
+ Sb$(Y)r(Zlrz)dv
BIG) + iLs~(z)l(x) L)dx
(3)
An analytical form can be derived for the entrance formula, sl(z); values of the exit formula f(z,L) are obtained by the Monte Carlo technique. We have chosen the parameters T z = 300°K and T1 = 100°K to approximate conveniently room temperature and the boiling point of liquid nitrogen, L/d = 1, 10, and 50, and finally €0 = 900 for all lengths and 2100 for length 10. The degree of specular reflection as defined below is abnormally high with these values of eo, but the computer time required is shortened and the over-all effect of eo more firmly established. The Entrance Formula. For that part of the problem which is to be solved analytically we are able to follow closely the procedure developed by DeMarcus'O for the case of a constant probability of specular reflection, [ P ( E )= constant]. Therefore we omit much of the detail of the derivation. Consider a beam of molecules at temperature T striking the wall of the capillary. The fraction which undergoes n successive specular reflections is, by our model
J
P(e)nF(e)de =
r1r2r3 *
0
r,
=
nT/(nT
+
*
- rn
eo)
+
We define rl = 1 / ( 1 eo/T)as the degree of specular reflection for the sake of discussion. Due to the nature of our model, the function r ( y , z)is symmetrical about y in spite of the temperature gradient aIong the capillary, i.e.
r(Y,Y + a)
=
r(Y,Y - 4
Following DeMarcus'O it can be shown that The Journal of Physical Chemistry
where
K(y, z) = lim r(9, z) fO+W
and r, is characterized by the temperature of the wall a t y. Now consider the equilibrium situation: I n Figure 1, set PI = P2 and TI = Tz. Using the well-known properties of an equilibrium gas and Clausing's integral equation for the collision density, we arrive at the formula s1(z) = (1
~
- r1)[(1- r1)121(x) + ( -1 4nl(z/2) + r l d l
+ ...1
- dnl(z/3>
where
n l ( z ) = lim sl(x) = m
f0'
-[ + dZ)"* + (x2 2 d (z2
52 +
d2)l/2
- 2x1
and now r, is characterized by the temperature of the reservoir at the entrance. This formula holds equally well if there is a temperature gradient along the capillary, again due to the nature of our model. Similarly, we can derive the formula
&(x)
+
- r l ) [ ( l - rl)Nl(s) 2r1(l - rz)Nl(x/2) 3rlrz(l - r3)N1(z/3) = (1
+
+ ...I
where
~ l ( z= ) lim ~l(z)= to+m
4 d2
-[222
- 2x(z2
+
d2)'/'
+
d2]
The Exit Formula. The calculation outlined below was performed on a Burroughs 5500 digital computer. Random numbers, R, where generated with a power residue formula (sometimes called the multiplicative congruential method), R,+l = CR, (modulo 8 9 , where C = 541755813883 and Ro = 1 in the decimal number system, and the computer was used a t double its normal capacity of 13 octal digits through a special routine. The R, are considered to be integers in the generating formula and an additional division by 813 is required to normalize them to the unit interval. This particular sequence has a period of Z3'. Figure 2 shows the coordinate system of a molecule leaving the surface of the capillary. The distance
KNUDSEN LIMITINGLAWOF THERMAL TRANSPIRATION
ck NORMAL
D I R E C T I O N OF CAPILLARY A X I S
I,'
'\
AX
,,,,.
4i
%
-a' ,' ,'
I
Figure 2.
Coordinates of a molecule leaving the capillary wall.
traveled down the tube between successive wall collisions is
Ax
=
sin 8 cos 0 cos p 1 - sin2 0 cos2 cp
It should be noted that in our program the angles 0 and p were always chosen successively in that order. Since the random number generator does not really produce random numbers, one of the severest tests to which it can be put is that successive values of R appear to be random for the purposes of the problem being solved. Statistical tests have shown that the power residue method scores well in general on this point. Nevertheless, the coupling between 0 and p could very well have produced a small bias in favor of one direction down the capillary. Therefore we introduced the following artifice to ensure that '5-andom" values of Ax were symmetrically distributed about Ax = 0. The angle p was limited to the range [0, a/2] and a new random number was chosen to decide whether Ax was to be positive or negative. The temperature gradient along the capillary was taken to be constant. The various random variables were related to the uniformly distributed random numbers, R, by integrating the corresponding distribution function. Let r be a random variable which follows a distribution function, P ( r ) , over the range 0 < r < a. Then each value of T is obtained by the relation
3877
Let x be the starting position. Choose four random numbers: R1= [0, 11 = sin 0, R2 = [0,a/2] = p, R3 = [O, 13, and R4 = [0, 11 = e-"T, where T = 300 (200/L)x. Calculate /Ax/ by eq 4; if R8 < 1/2, then Ax = -1Ax1, otherwise Ax = /Ax[. Calculate B from R4 and choose RE,Re, . . . R, = [0, 11 until R , < e-"". The new starting position or next diffuse reflection is x (n - 4)As. The calculation ends when a diffuse reflection occurs at a position greater then L or less than zero.
+
Results The results of the Monte Carlo runs are summarized in Table I. Values of f(x, L ) are given by the ratio of the number of molecules exiting to the number of trials or histories generated. Transmission probabilities were calculated for both directions, the integral in eq 3 being evaluated graphically. For the reverse
Table I: Monte Carlo Results for the Exit Function; Entrance a t 300"K, Exit a t 100°K
Lld = 1 €0 = 900 500 trials per position L/d = 10 €0 = 900 1000 trials
Starting position
exiting
0 0.2 0.4 0.6 0.8 1.0
126 168 214 290 354 385
Lld
65 177 547 866 953
Lld
Starting position
No.
0 1 5 9 10
10 2100 1000 trials =
eo =
50 900 1000 trials =
EO =
P rn
R,
=
0 1 5 9
No. exiting
10
49 147 512 859 963
0 0.4 1 4 10 25 40 46 49 50
12 27 47 96 206 468 780 892 969 991
JoP(t)dt.
where the R, are adjusted to fall in a range determined by 0 < r < a. The exit function was evaluated by starting a molecule a t a chosen position a t the wall with a diffuse bounce. The positions chosen tended to be clumped near either end, where the exit function is of greatest importance in determining the transmission probability. A brief outline of the generation of a molecular history is given here. The adjusted intervals of the random numbers are given in brackets; the natural interval of the random numbers as generated by the power residue formula is always [0,1].
direction, cold to hot, the definitions of exit and entrance were interchanged. Transmission probabilities were also calculated for the case of no temperature gradient using the variational solution of Dehlarcuslo
Q
= 8(
ABL - AC - B2 4AL2 - C
+ &(L)
A
=
1
B
=
lL&(x)dx
Volume 70,Number 18 December 1966
3878
GEORGE A. MILLERAND RALPHL. BUICE,JR.
C = LLzS,(x)dz
Table I1 : Transmission Probabilities for Free Molecular Flow
For our model the above integrals become
B
=
(1 - ?-l)[(l - r1)I(z) 4rl(l
- ri)I(z/2)
+
+ gTlrz(1 - T3)1(%/3) + . . . ]
I(z/n) = j - p l ( z ) d z
c = (1 - r1)[(l - r1)J(z) 8rl(l - rz)J(z/2)
+
+ 27Tlrz(l - To)J(z/3) + . . . ]
J(z/n) =
LL/
ZNl(z)dx
For the special case of no specular reflection, Le.,eo + 00, the calculations have been perfonned elsewhere;12 Q is then independent of temperature. Figure 3 shows the exit function for the three capillaries with the entrance (z = 0) a t the hot end. The curve for eo = 2100 and L/d = 10 is omitted. For the case of no temperature gradient or of only diffuse reflection the exit function would obey the relations f(Ll.2, L ) =
f(x, L )
+f(L-
Capillary length t o diam ratio,
L/d
Specular parameter,
Transmission probability,
Std deviation,
€a
Q
C
...
m
300 100 100 300
900 900 900 900
0.514 0.695 0.719 0.558 0.539 0.109 0.142 0.140 0.118 0.113 0.232 0.245 0.133 0.123 0,0253 0.0691 0.0653 0.0322 0.0300
Exact Exact 0.007 Exact 0,008 Exact Exact 0.010 Exact 0,009 Exact 0.011 Exact 0.009 Exact Exact 0,0044 Exact 0,0040
Entranoe temp, OK
Exit temp, OK
... 300 300 100 100
1 1 1 1 1 10 10 10 10 10 10
10 10 10 50 50 50 50 50
...
...
m
300 300 100 100 300 300 100 100
300 100 100 300 300 100 100 300
2100 2100 2100 2100 900 900 900 900
...
...
300 300 100 100
300 100 100 300
m
900 900 900 900
‘/z
2,
L)
=
1
As can be seen, perhaps better from the data in Table I than from the graph, the presence of a temperature gradient results in a detectable deviation from these relations. The effect is of minor importance however. A summary of transmission probabilities is given in Table 11. The variational solutions are listed as “exact,” whereas in fact they are known only to be much more accurate than our Monte Carlo calculations. The standard deviation of the Monte Carlo values of f(x, L ) for N trials is given by
0
x/L
I
Figure 3. The exit function for various capillary length to diameter ratios with the critical energy of specular reflection equal to 900.
All of the uncertainty in the calculation of Q came from evaluating the integral in eq 3. The trapazoidal rule was used to weight the various data points in determining the standard deviation. For example, for L = 10, the standard deviation of Q was given by
from the Knudsen limiting law increases as anticipated (Figure 4). Second, the existence of a temperature gradient along the tube has only a secondary influence on the transmission probability, the important variable being the reservoir temperature (Figure 5 ) . Apparently the decisive event is the initial specular reflection of an u [(1/4)uo2 -k (25/4)ui2 f entering molecule which places its first diffuse reflection 16~5‘ (25/4)09‘ ( 1 / 4 ) ~ i o ~ ] ” ~farther down the tube, where the exit function has a larger value. Mathematically stated, the functions where u0 is the standard deviation of sl(O)f(O, L ) , etc. s1(z) and Sl(z), which are dependent on the reservoir I n examining the results two points stand out. temperature, essentially determine the temperature First, as the degree of specular reflection increases,
+
+
the values of the forward and reverse transmission probabilities diverge more and more and the deviation The Journal of Physical Chemistry
(12) L. M.
Lund and A. S. Berman, J . AppE. Phys., 37, 2489 (1966).
3879
KNUDSEN LIMITINGLAWOF THERMAL TRANSPIRATION
forward and backward values of Q should 9 pendent of the effect in the first order.
Q.20/ IOO'K ,
.40 IOOO/G'
.so
I .20 "
Figure 4. Capillary transmission probabilities for the entrance a t 300°K: (upper curve) and 100°K (lower curve) as a function of the critical energy of specular reflection. Solid curves are for no temperature gradient. Circles are Monte Carlo values for a capillary with a uniform temperature gradient between 300 and 100°K.
Q 15
Figure 5. Unified plot of capillary transmission probabilities as a function of the degree of specular reflection a t the entrance temperature. Solid curve is for no temperature gradient. Circles are Monte Carlo values.
dependence of Q,whereby QlZ < Q21, and the exit function plays only a minor role in this respect. Within the limits of our model this leads to the important conclusion that free molecular flow experiments performed under differing, uniform temperature conditions should reveal the same order effects observed by Hobson and co-workers in connection with the limiting law. Before discussing capillary flow data, however, it is necessary to deal with the problem of back scattering. Transmission probabilities may be decreased by artifically roughening the capillary wall.13 Due to a sort of "roof top" effect,14 molecules impinging at low angles are preferentially scattered backward. The effect appears to be important on normal surfaces as well. We have assumed that back scattering causes a small and equal diminishing of the transmission probabilities in either direction. The effect of temperature on Q is to be attributed as before to the temperature sensitive degree of specular reflection. Our calculated values of Q are too large by whatever amount must be subtracted for back scattering, but the ratios of the
be inde-
Lund and Berman15pl6 have made precise measurements of the gaseous flow in metal and ionic crystal capillaries a t low pressures. Although their data cover a temperature range of only 0 to ZOO,transmission probabilities could be correlated in terms of the Lennard-Jones gas interaction parameter over a much wider range of effective or reduced temperature. ~ Taking arbitrary pairs of reduced temperatures which correspond to TZ = 300°K and TI = 100"K, we find that the ratio of transmission probabilities is always close to unity Qz1/&1z
S &zz/&11
= 1.02 to 1.03
The correlation included the light gases hydrogen and neon, but the behavior of helium was anomalous and no conclusion can be drawn about this gas. Eschbach, Jaeckel, and Muller" have measured transmission probabilities of helium through a glass capillary over a very wide temperature range, -200 to +600°. Within the *5yo uncertainty of their results they observed no temperature dependence of &. The evidence from flow experiments, then, is that the ratio &21/&12 should be within a few per cent of unity between room temperature and liquid nitrogen tempem ture and that the Knudsen limiting law should be obeyed within the same limits. Nevertheless the extensive TPD measurements of Hobson and co-morkers4p5 have clearly established larger deviations from the Knudsen limiting law (Table 111). Also the deviations decrease with increasing capillary length, whereas our calculations show the opposite trend. It would be interesting to know if these apparent contradictions lie in the assumptions of our model, including possible entrance effects caused by a non-Maxmellian distribution in the reservoirs. The answer might well come from further molecular beam work, but such studies are difficult to carry out, particularly at angles involving back scattering. We suggest that simple flow experiments would be quite informative. For example, it would be very helpful to have precise data on free molecular flow through glass capillaries (1) under isothermal conditions, but over a wide temperature range, and (2) under positive and negative ~~
(13) D. H. Davis, L. L. Levenson, and N. Milleron, J . Appl. Phys., 35, 529 (1964). (14) W. C. DeMarcus, U. S. AEC Report K-1435, 1959. (15) A. S. Berman and L. M. Lund, Proc. Intern. Conf. Peaceful Uses At. Energy, Bnd, Geneoa, Sept 1968, 4, 359 (1959). (16) L. hl. Lund and A. S. Berman, J. Appl. Phys., 37, 2496 (1966). (171 H. L. Eschbach, R. Jaeckel, and D. Muller, Trans. A'atZ. Vacuum Symp., 8, 1110 (1961).
Volume 70,Number 18 December 1966
GEORGEA. MILLERAND RALPHL. BUICE,JR.
3880
Table I11 : Neon Transmission Probability Ratios for Pyrex Tubing" L/db
Qn/Qn
5.8 8.0 11 24 210
1.23 1.23 1.15
1.14 1.05
Based on thermal transpiration data from ref 4 and 5 ; The distance over which the temperature gradient takes place is much smaller.
T2 = 295"K, TI = 77.4"K.
'
temperature gradients. Deviation from the Knudsen limiting law should be established by TPD measurements on the same capillaries. The effect of grinding
The J o u T of~ Physical ~ ~ Chemistry
or etching the capillary wall might throw further light on the role of back scattering and also might prove to be a practical method of attaining Knudsen limiting law behavior. One attractive feature of flow measurements is that only relative pressures are needed since the logarithmic decay of pressure is determined. The manometer may be operated at the reservoir temperature without the need for calibration, provided the response is known to be linear. I n the measurement of TPD, on the other hand, there are two reservoirs at two different temperatures, which presents a basically different problem in pressure measurement.
Acknowledgment. This research was supported in part by National Aeronautics and Space Administration Grant NsG-657.
