A SD-TDR- b3- 7542fPart lil
MOLECULAR FLOW AND THE EFFUSION PROCESS IN THEMEASURDIENT #01 VAPOR PRESSURES
Robert D. FreemanOklahoma Statc University
TECHNiICAL DOCUMIENTARY REPORT ASD--TDR-63-754, PL. IIINovember, 1967
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I• • I01 --... ... Dk~jk,%V
MOLECUL-AR FLOW AND THE EFFJSION PROCESS IN THEMEkSURIMENT OkE VAPOR PRESSU"S
Robert D. Freeman
This document has been approved for public releaseand sale; its distribution is unlimfted.
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FOREWORD
This report was prepared by the Research Foundation and theDepartment of Chemistry, Vkidhouma State University, Stillwater, Oklahoma,under USAF Contract AF 33(657)-8767, This contract was initiated underProject No. 7360, "The Chetnistry and Physics of Materials", Task No.736004, "Special Problems in Katerials Physics". The work was adminis-tered under the direction of the Air Force Materials Laboratory, Researchand Technology Division, with Mr. Paul W. Dimiduk, MAYT, as projectengineer. This report was submitted in August 1967 for publication.
This report is an account of the research accomplished betweenI September 1964 and 31 Niay 1967.
Thý author wislIcs to acknowledge the many contributions of thegraduate sutdents ai.d research associates who have been associated withthis research program, and who should be credited with co-authorship ofthe various scctiorb. They are E. A. Elphingstone (section V), R. E.Gebelt (sections IV and V), J, G. Edwards (section II), and Ruth C. Erbar(section III). The exccllent craftsmanship of the machinists and instrumentmakers in our deparcmental Machine and Instrument Shop has been invaluableznd is gratefully acknowledged.
This technical documentary report has been reviewed and is approved.
Chief
Thermo and Chemical Physics BranchMaterials Physics DivisionAir Force Materials Laboratory
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iItAi ABSTRACT
Our extensions to conical orificts of Clausing's analysis of angular
distribution of molecules effusing from cylindrical orifices has resulted
in numerical values for transmission probabilities and recoil-force correction
factors which are tabulated. With these results, it is demonstratcd that
the optimum orifice geometry for (1) recoil force measurements is a diverging
conical orifice with semi-apex angle of 30', (2) delivery of maximum fraction
of effusing molecules onto (or into) a target (aperture) is a long cylindrical
orifice.
Modifications to the angular distribution apparatus are described and
experimental results given for four orifices and two gases over the pressure
range 5 to 900 dyn/cm2. The most interestng aspect oi the results, one
apparently not previously noticed in angular distribution results, is the
presence of maxima and minima in plots of A vs. 0, where A is the (experimental
value - theoretical value) of Pe, the fraction of effusing molecules which
flow per steradian at angle 0 from the orifice axis. Vhese maxima and minima
have been correlated with the relative contribution from the orifice wall
to the total flux at angle 0.
Addltolual Lesuits for experimental transmission probabilitics of orifices
determined by the Multicell technique are generally in agreement with theo-
retical values within 2 to 5%.
The Riker technique for simultaneous determination of vapor pressure by
rate of effusion and by recoil force measurements has been refined to the point
that recoil force data are as repro'-cible as rate of effusion measurements.
Several sources of spurious recoil force have beer identified and eli.inated.
A new furnace and a ;aodificd automatic control system for the microbalance
are described.
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TABLE OF CONTENTS
Section
I. INTR DIITION . . . . . . . . . . . . . . . . . . . . . . . . . I
II. THEMRE•TICAL ANALYSIS OF MOLECULAR FLOW THROUGHCONICAL ORIFICES ............ ........................ 1
Ill. MEASUR1EMNT OF ANGULAR DISTRIBUTION OF MOLECULARFLOW THROUGH CONICAL ORIFICES ..... .............. .. 19
A. Experimental Apparatus ........ ................. ... 19
B. Experimental Results ............ .................. 24
C. Discussion ................ ....................... 33
IV. THE MULTICELL TECHNIQUE FOR EXPERIMENTAL DETERMINATIONOF TRANSMISSION PROBABILITIES FOR MOLECULAR FLOWTHROUGH CONICAL ORIFICES ............ ................. 57
V. THE MIKER TECHNIQUE .................... 67
A. Modifications to Apparatus ............ .............. 67
B. Experimental Results ........ .................. ... 71
VI. CALORIMETRIC STUDIES OF VAPORIZATION PROCESSES ... ....... 7I
REFERENCES ......................... ............................ 76
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ILIST OF ILLUSTRATIONS
Figure Page
1. Conical OrLifices and Parameters: A, DivergingConfiguration, T positive; Bi, Converging Con-figuration, T negative .................. ...................... 3
2. The Transmiseion Probability W as a functionof log (L/ro) at various values of T ..... .................. 7
3. The Recoil Force Correction f as a functionof log (L/ro) at various values of T ...... .................
4. Collimating effect of various orifice geometrieson effusing molecules ............. ...................... ... 4
5. CollmatLing effect of various orifice geometrieson effusing molecules ............. ...................... .. 15
6. Optimization of orifice geometry for recoil force ..... ......... 17
7. Diagram of the Experimental Apparatus ....... ............... 21
8. Experimental angular distribution data(Normalized beam intensity) for Nitrogenthrou h orifice 1 (T = 25.650 and
(1/,, •.....)......................... 2)
9. Ex erimertal [1(e)] and Normalized TheoreticalLQnCoso] Angular Distribution Data for T - 0.0O',L/rm , 2.44 ................... ........................... .. 27
10. Experimental [I(G)] and Normalized TheoreticalLQnCose] Angular Distribution Data for T - 30',L/rm - 2.C, with Nitrogen ........... .................... .. 28
Normalized Experimental LI(C)] and Thuoretical [QnCos0]Angular Distribution Data for Nitrous Oxide throughOrifice 2 (T - 8.5000, L/rm 1 10.08) ..... ............... 29
12. Normalized Experimental [I(e)] and Theoretical EQnCOSO]An ular Distribution Data for Nitrogen through Orifice5 (T -58.93°, L/rm - 11.01) .................. 30
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Figure Page
15. Norm'alized Experimental [1(0) ] and Theoretical [QnCOsC ]Angular DistIi bmt iou Dat a for Nitrous Oxide thuoughOrifice 3 (T - -58.95', L/rit . .i.O.) . .......... . 31
), . Normalized ExperitIMntal [1(0) ] and Ti•eoretical [QnCos6]Angular Distribution Data for NiLrogen through Orifice1 (T - 25.050, L/rm , 4.010) ......... ....... 52
25. f(A) plots for 120 aerieE ..................... 38
10. 6 plots for 120 aeries . . . . . . . . . . . . . . . . . . . . . . . 39
17. f(L) plots for 230 series . . . . . . . . . . . . . . . . . . .. . 4o
18. f(L) plots for 320 SerIes .................. . 41
19. f(W) plots for 350 series ..................... 42
20. f(6) plots for 420 series . . . . . . . . . . . . . . . . . . . . . 43
21. 62 plot (cf. equation 2u) for Run 125. Comparewith Run ll25 in Figule, 5 ............... .................... 44
22. The Critical Angles and the Angular Ranges fora Conical Orifice ................. ........................ 45
25. A plot for K. C. Wang's angular distribution data 27
for orifice with Tf - 0.00 and L/ro - o.954 .... ............... 418
2h. A plot for K, C. Wang's angular distribution data2n
for orifice with T 0.0' and L/ro = 2.59 .... ............ 49
- ... J . I,,. .. .. OiL ...... 52
26. Normalized projected area of inside wall of Orifice 4 ......... ... 53
2'[. Niralied projected area of inside wall of Orifice 2.................54
28. Normalized projected area of inside wall of Orifice 3 ... ........ 55
29. Weight loss through orifices with various (L/D) ratiosvs. theoretical transmission probability ............ .. 62
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LIST OF TABLES
Tabble Page
1. Transmissioii rrobsbllitlcs and Recoil ForceCorecttouns for Conical Orifices ............. ................. 4
2. Orifices, Gases, and Pressurvs Used inAngular l)istribution Studied. .......... ................... ... 26
3. Cotapari.so ol Expcrimental and TheoreticalTiansmisiion PlobabilitiCs: Run '7 .......... ................ 61
l. Experimental TranEmission Probabilities forEight Cylindrical Orifices (Set ) ..................... 63
5. Experimental Transmission Probabilities forEight Cylindrical Orifices (Set II) .............. ............... 64
6. Experimental Transmission Probabilities forEight Conical Orifices (Set III) ......... ................. ... 65
7. Miker Data for Vaporization of Tin ....... ............... ..... 72
8. Identification of Major Comunercial Componentsof Apparatus ................. ........................... ... 76
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LIST OF S(Y!MOLSDe fined]
Symbol o n Page Definition or Explanation
AL 51. projection at angle e of arca of orifice exit.
A 5] projection at angle 8 of area of orifice entrance.
51 projection at angle 6 of orifice wall area visiblefrom molecular beam detector.
AE 51 projection at angle e of the area A (1 0; DL ); seep. 51.
A(1)0;D L) 51 area of overlap cf circles rTr 2 , and Trr1 pro-0jected at angle 6 onto the plane of 1717'
•~--o
a. 58 cross sectrona) area of ith orifice.
a 16 integration limit.n 2
a 11 cross sectional area of orifice entrance, •r
o 0
b 60 constant in equation for straight line.
b 16 integration limit,n
D 58 diameter of orifice.
dx differential of x; other differentials are listedunder the function differentiated,
exp designates experimental value.
F 18 force exerted by molecules impinging on targetsubtending angle y.
F 18 force exerted by all molecules impinging on target,o from an ideal orifice,
F(LI/D,T) 59 function of L/D and T.
f 2 recoil force correction factor.
4 recoil force correction factor for convergingconical orifice.
f 18 angular recoil force correction factor.
dfr 10 incremental recoil force correction factor,
f(A) 36 the fractional difference (Px-P )/_ .
gi 58 total mass effusing from cell in time T.
do 58 incremental effusing mass.
298 66 standard enthalpy change.
I 58 represents value of an integral.
33 measured ion current at angle e, ++2 33 relative molecular beam intensity, 1 /.+
6 e wi index.
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0. 58 mass effusing per u it area or ifice in t tine "'rI
159 mass effusing per unit area of M i Irice in Lime C I -
Ce 1St ed f uron I ensi t- squares I itc thirough Lcx p utIie topoi lnt s6
J .59 nlmass cllusintu per- unit a112a 01 ideal o)if ice int iic ".
K. 58 exp1rijental transmission probabi i t\.
K 59 experimenta 1 transmission probability for ideal0orifice.
Kp 60 experimental transmission probability, fromi experimental points.
K1 60 experimental transmission probability, fromleast-squaces line through experimental points.
K 60 experimeýntal transmiission probability, average.
L 2 length of orifice along its axis.
D 58 molecular weight.
rn 58 the number of cells in a mu Iticcll experitment
"1 60 constant ill equation for straight ine.
Ny 12 number of molecules which effuse per second into
a cone with semi-apex angle y
N 12 number of molecules which enter the orif ice through00
o ~aao per second.
NL 12 number of molecules whi ch effuseL from the orificeexit per second.
dN 19 number of molecules per second which pass throughthe incremental solid angle dL.
dN (L) 33 number of molecuILcs which effuse per second into-iic elid aI I e at:.
dNe (L) 33 number of me lc cu l s wihich effuse pIer second intothe solid angle I 2"' Sia3 A.
N, (L) 34 numbCr of molecules whichi effuse per second intothte finite, but small, solid angle 2.
p 58 pressure in effusioln CelI.
P 72 presoure iii e inus ion cell de ermined by effusionn1t aisureVnli, t .
P 72 plessu1e iD effusio( ceCI d et Crmi njd by r'ec0iforce measureme.Lnt.
35 probab i liiy density funt ion for effusing mo lec les.
35 P0 , cxperimenita 1.
i y,
I
p 35 P , thcolettical,
P 35 F6, experimenta I, at J =L1o6
Q 33 aL)gula1 dist ribut ion funct ion; Q =1 for idealor if ice
Qn 12 7Q .
12 Q with n1=
1. 58 gas constant
r 2 radius of int rance to orifice.
r 3 radius of exit from orifice.
r 4 radius of smaller end of orifice.m
T 2 angle between orifice axis and orifice wall,measured in a plane which contains the orificeaxis .
T)(0 10 Clausing's angular distribution function.
th designates theoretical value.
dt 58 differential of time.
W 2 transmission probability (theoretical).
W 12 angular transmission probability.
dW 12 incremental transmission probability.
y 12 angle.
A 36 the difference (P2-P t)
AH298 66 standard enthalpy change.
6,5 36 functions of P x and I1' 26 3 angle, especially the angle between a molecular
trajectory and the orifice axis.
o 58 temperture, absolute (section IV only).
6* 37 6*=arctan (r +rL)/.L; see Figu/e 1,
kao 33 molecular flux incident on orifice entrance.
1 33 index.
TT 3.14159
"" 58 interval of time.
oi 16 total recoil force for effusion from ideal orifice.
do 16 incremental recoil force.
V(x) I0 normalized molecular flux on orifice walls.
2 d 34 solid angle subtended at orifice by molecular beamdetector aperture.
subscript; refers to a solid angle.
& 12 incremental solid angle.
x
Si-CTION I
INTRODUCTION
The Knudsen effusion techniquei, 2 is widely used, in varied guises,
to obtain vapor and/or dissociation pressure data, especially at high
temperatures. In actual laboratory operation the conditions under which
effusion occurs are rarely, if ever, the ideal conditions assumed in
the derivation of the simple Knudsen equation. This report summarizes
work on several approaches designed to clarify the understanding of the
effusion process under non-ideal conditions. More detailed introductory
paragraphs are included in each of the following sections.
SECTION II
THEORETICAL ANALYSIS OF MOLECULAR FLOWTHROUGH CONICAL ORIFICES
Vapor pressures and the composition of vapors at high temperatures
are oftpn determined by effusion techniques 3, e.g., Knudsen effusion,
torsion-Knudsen-effusion-recoil, and target-collection methods; furthermore,
mass spectrometry of high temperature vapors is often facilitated by the
use of an effusion cell as the vapor source 3 . Derivation of thermodynam-
ically significant data from the results of effusion experiments requires
knowledge of the relation between the measured quantities and the equilib-
rium pressure in the cell; this relation must take into account, among
other factors, the non-ideality 4 of the geometry of physically-realizable
effusion orifices.
Tihe effect of o .:ifice ge y',';h rate. Of -oleCUlZI effusion
throngh the orifice has bccu analyzed by several investigaLors5- 0 , whose.-r'sults have usually been given as nmoerical values for the transmissionI
probability, i.e., the probability that. a molecule which enters the orifice
through one end will exit from the opposite end. The effect of orifice
geometry upon the angular distribution of molecules effusing from an
orifice has received less attention but an analysist m for cylindrical
orifices has been made and the results have been used to obtain the
"recoil-force correction factors" applicable to torsion-Knudsen-effusion-
recoil measurements 3 .
In Part I12 of this report and in a conLribution to "Condensation and
Evaporation of Solids"'13 (subseqoLntly cited as CES) we hive presented
an anialysis of the flow of rarefied gabes through and from conical orifices.
Equations were derived for calculation of the incident molecular flun:
density along the orifice walls, the transmission probability of the orifice,•
the angular uistribution of molecules leaving the orifice, and various
other functions. In Part I, but not in CES, the results of extensive com-
putatioas of the various functions were reported; for convenience, one
table and two Figures from Part I are repeated herein, and the results
discussed briefly. Subsequently, these results are compared with other
theoretical results; comparison with experimental results is deferred
to Section III C and IV of this report.
In the last portion of this Section we shall exploit the angular
distribution functions for conical orifices, which have not previously
been available, to consider the determination of an optimum orifice geometry
(within the class of right circular cones) for particular experimental
configurations.
A. RESULTS OF THE NUMERICAL CALCULATIONS
in Figure 1 the various critical orifice parameters are defined by
illustration. Values of the transmission probability E, and the recoil-
force correction factor f are given in Table 1, and in Figures 2 and 3,
respectively, for conical orifices with O(L_/ro)f1O and with -90!CT3+O9,
i.e., for both converging (_1 negative) &nd diverging (T positive) conical
orifices. Transmission probabilities are given only for the diverging
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rL FLOW
Figure 2. Conical Orifices and ParamcLcrs: A, DivcrgIngConfiguration,, T positive; BConvc-rgivtg Conliguration,,
T negative.
TABLE I
TRANSMISSION PROBABILITIES AND RECOIL FORCE CORRECTIONS
FOR CONICAL ORIFICES
T (L/r ) a w f (f*/r 2)am _ _ __ _ _m
0 0,1 0,952399 0.9683220.2 0.909215 0.937308
0.4 0.82408 0.87847
0.6 0.77115 0.82471
0,8 0.71778 0.77620
1.0 0,67198 0,732692,0 0.5142 0.57254.0 0,3566 0.4024
6.0 0.2754 0,31258.0 0.2253 0.2564
10.0 0.1909 0.2177
100 0.1 0.967347 0.986104 0,9803300.2 0,93835 0.97200 0.961270,4 0.88938 0.94451 0.92584
0.6 0.84990 0.91904 0.89445
0.8 0.8176 0.8960 0.86701.0 0.7908 0.8756 0.8432
.20 0.7058 0.8035 0.7621
4. 0, C 0.730- 01.6899
6.0 0.6051 0.7107 0.65898.0 0,5895 0.6970 0.6425
10.0 0.5802 0,6892 0.6328
200 0.1 0.978646 0,999546 0.988540
0.2 0.96027 0.99816 0.97764
0.4 0.93057 0.99407 0.95804
0.6 0.90793 0,98941 0.94150
0.8 0.89034 0.98492 0.92770
1.0 0.87642 0.98087 0.91625
2.0 0.8370 0.9682 0.8813
4.0 0.8108 0.9633 0.8564
6.0 0.8022 0.9660 0.8480
8.0 0.7984 0.9699 0.8442
10.0 0.7963 0.9736 0.C421
14• • n m • •
T (_L/r_) a _(_ f/r_ 2)a
Im30 0,1 0.986915 1.008954 0.9680,2 0.976141 1 .016307 0.9881980.4 0.95973 1 .02782 0.978480.6 0.94812 1 .03652 0.970790.8 0.9396 1.0434 0.96488.0 0.93338 1 .04919 0.96004
12.0 0.9177 1.0687 0.94734.0 0.9095 1.0908 0.94016.0 0.9073 1.1044 0.93818.0 0.97065 1..1140 0.9373
10.0 0.9060 1.01212 0.9369
40 0.1 0.992680 1.014589 0.9970830.2 0.987008 1.026941 0.9944940.4 0.97902 1.04683 0.99034.06 0.97389 1.06220 0.98732.0 0.97044 1 .07444 0.985131 0 ( 0.96806 1 0844 6 0.98352
2.0 0.96288 1 11632 0.97974440 0.9607 1.1462 0997806.0 0.990381 .1618 0.97768.0 0.9601 1,1720 0.9775i0.0 0.9599 1,1793 0,9774
45° 0.1 0. 994775 1 .016077 0. 9981040. 2 0. 990881 1 .029628 0. 9961.o80.4 0.985662 1 .051287 0.9939490.6 0.98249 1.06782 0.992210.8 0.98047 1 .08086 0.99101i .( 0' f'-;912 I .0914 3 (j. , e o 12.0 0.9764 1.1242 0.98834.0 0.9754 1.1536 0.98756.0 0.9752 1.1664 0.98748.0 0.9751 1.i778 0.9873
10.0 0.9750 1.184 4 0.9873
50° 0.1 0.990420 1 .016729 0.9988310.2 0.993877 1 .030084 0.9978550.4 0.99060 1 .05267 0.996420.6 0.98863 1 .0t918 0.995490.8 0.9d772 1 .(08202 0.9q4t w1.0 0.98701 1 .0i2 31 0.99/4472.0 0.98570 1.12355 0.993634.0 0.9,353 1I .1504 0.99336.0 0.98',2 1 . 1636 0,99338.0 0.9851 1 .1717 0.9932
10.0 0.9,b l 1. , 74 0. 9 . 2
T1 (L/rm)a m ,(U/r 2)
600 0,1 0.996591 1.015712 0.99913440.2 0.997715 1.028324 0.999374
0.4 0.996761 10.47307 0.9990260.6 0.99L304 1.060889 0.998831
0.8 0.9960u 1.07108 0.998731.0 0.99592 1.07904 0 .9866
2.0 0.99571 1.10207 0.998544.0 0.9957 1.1205 0.99856.0 0.9957 1.1289 0.99858.0 0.9956 1.1338 0.9985
10.0 0.9956 1.1372 0.9985
700 0.1 0.999631 1.011961 0.999936
0.2 0.999450 1.020864 0.9998950.4 0.999296 1.033229 0.999853
0.6 0.99924 1 .C'4141 0.99983
0.8 0.99921 1,04724 0.99982
1.0 0.99920 1.05160 0,99982
2.0 0.9992 1.0b35 0.99984.0 0.9992 1.0721 0.9998
800 0.1 0.999966 1.006048 0.999999
0.2 0.999957 1.009755 0.9999970.4 0.999951 1.014072 0.9999960.6 0.999Y5 1.01051 1.000000.8 0.99995 1.01809 1.00000
1.0 0.99995 1.01919 1.00000
2.0 1.00O 1.0219 1.0000
4.0 1.0000 1.0236 1.0000
a. r = radius of smaller end of orifice. (L/r m2) i8 the appropriate
recoil force correction for Convergin orifices when (L/!) for the
orifice is calculated with r rather than ro (see Figure 1 and
references 12 and 11).
d
1.0 -- 60°
40
0.9- 3Q0
250w
220
0.8 200
0.7 150
-1,0 OX) 1.0Log (L/ro)
Figuie 2. The Transmission Piobability W as a fun'ti-,n oflog (L/ro) at various values of 1'. 7i
U
1.20450
5Q0
7
115 /����600
/// 7300
HO /7 700
250
1.05
� 220
[00
200
0.95
0.90
� 50 100-1.0 0 1.0
Log (L/r0)
I igure �. The Rec�i1 Force Crrecti �n a� a func tio1 �g (i/�>) at various vaJ.u�s oL T.
8
configuration because, for a given o0. i flCe geomt ry, W is thc iwme fox gas
flow in either directions: 14. The loguritLULic scale on ihc abscissa ofFigures 2 and 3 is ,sed only to siwplify picscntaticn of the results.
Table ) and Figure 2 reveal no unexpected va):, atiOi of W; it decreases
with increasing / and increases ,s LI Increases, as would be predicted
intuitively. However, Figure 3 reveals an unexpected variation of f with
T: for T 5 L2'° f is ,e~ate, than unitv. Hence, the recoil force generated
by molecular effusion irow a conical orifice with T > 220 is greater than
that from the corresponding 4 ideal orifice. This result arises because
(W) the average angle at which molecules effuse from conical orifices is
smaller than for the idt:al oritice, and (2) as T increases above 200, the
transmissloni probability rapidly approaches unity; the momental component
along the coUical olifice axis is therefore greater than that along the
ideal orifice axis and the recoil force is greater. This "focusing"
effect of coi:ical orifices is a uiaximupm at -- %°, the precis-- 'alue depending
Oi (!jrn/':o).
NuraeriVel values for various other quantities, e.g., the incident
molecular flux on the erifice wall and the angular distribution of effusing
molecules, have been calculated but are not presented here; an extensive
tabulation of these functions is available elsewhere 4.
B. COWPARISON WITH OII UR THEORETICAL RESULTS
It Was shown il, Part I and in CE'S that, where comparison was possible,
the rCsults presentcd were in agreement with those of other workers. In
particular, our r t..i.a are in excellent agreement with the transmission
probabilities of Iczbowski, et a].ý' for conical orifices, ar'd witli the
closely-bounded values for the transmission probabilities o,- cylindrick.l
orifices obtained by DeMarcus and Hlopper"a. In view of the different
approaheL, the independent verification et derivations, and the :onsider-
ably different numerical techniques used ii solving the integral eqaations,
by four groupsf 5 , P0, 1 , 14 there can be little doubt of the validity of
the numerical values for the transmi.sion probability for cylindrical and
conical orifices; these values apply rigorously only to the assumed model,
of course.
We know of no o.h. r nlys. s ol. zL11iiar distribution to- conical
orilices, and for cylindrical orifi-,,s only of ClaIsiUg's analysis-".
Clausing assumed that Lhe normalized incident density (or molecular flux)
on the walls of a cylindrical orifice could be expressed as a linear
function of the distance, along the orifice axis, from the entrance.
The accuracy of this approximation has been discussed by Edwardslo whe
has also shown that, with this spproximnation, Clausing 's distribution
functionas ~i T(O) may be derived in detail directly irom our more general
distribution junctions which apply to conical and cylindrical orifices;
it has also been possible to integrate in closed form the resulting
angular distribution equation foi __yjlindrical orifices.
One further argument may be advanced for the validit.y of our angular
distribution results. In solving the various equations 3 the first quantity
obtained is 4(x) the normalized incident density on the orifice walls;
these values of '(x) are then used to obtain W4 in a direct way. The same
values of ,(x) are used in the more Involved computation of k' by numerical
integration of the angular distribution functions. The two values of W
are in good agreement with each other and, as already noted, with tbC
values of Iczkowski, et al.6 which would seem to indicate that both ý(x)
and the angular distribution functions are correct.
It is difficult to compare our results with those of Davis, et al. 1X;
their paper lacks detail in both derivatt•.. and results. Our impiiession
is that their derivation is not rigorous. In any case the. re-.ulto shown
in their Figure 7 for the transmission probability of "ceonical nozzles"
appear to be higher than those of Table I by several percent for small
angles T and lower for large angles T.
Sparroo and Jonssorn•3 have also analyzed the mass flow through conical
orifices and have used the results to formulate analysis of energy transfer
between gas Oad orifice walls. If, in their equation (12) aad the ordinate
of their Figure 5, 12 is made zero, the resulting term is equivalrint to
our transmission probability W. The results in their Figure 5 appear to
be in good agreement with values of L% in 'fable 1, and !(/i') of their
figure (2) appear to agree with out •(x) 2. Sparrow and JOhkcSon obtain
a number of useful but seldom used relations (theii equations 6-11), VOse
of which we have a.so generated'. 3'-.
10
II
Richley and colleagues hav. iucently reported a sceies of analyses
of jolcul ar fow Hthrough fyl indri -a1 tubes "a C -converging and divergingjiob d4.tubes- aud slotsb, s and zbrHoadh cylindri cal iubys .iL &: contributIon
f from surfice diffusionwcV The investigation by Cook and kiehley of
angular distribution from cylindrical oriliecs is based otn Clausing 'S
analysis 1 0 which has already been ,orrr.latcd with the present results.
in their analysis 9b of flow through conve,:gieng anf diverging tubes and
slots, Riebhhy and Iceynolds obtain, by 'tterative solution of the appro-
priate Fredholm Integral equation, values for the normalized incident
density 4(x) on the orifice wall ( their f"lux ratio D-/a') and use these
values to obtain -lie flux distribut.ion evcer the plane of At! exit end of
the orifice; finally, thI exit plane flax distribution is integrated over
the exit area of the orifice to obtain the traunmiasuien probability E
"their Pt), The minor discrepancies which exist between their results
and oLajs appear to arise from (i) their use of an iterative solution for
r:.,/n. raLlier tItan cur 3,more di rec-t soltý eon 3, 14 for 4(x), and (2) their
introduction of an addiLional numerical integration (i.e., to evaluate
the exit plane flux distribution) in dhe camaputational sequence leading
to W, rather than calculation of U4 ditrectly from the 4(x) in one step1 3
C. OI'TIMIZATTON OF ORIFICE GEOMLETRY
An orifice geometry will be considered optimnem if, for any given rare~~~~~•.-1 -- .......... tIF r s eC t t )
of effuslon, the quantity. ben ".asnred is L,,it. ze wh respect to the
orifice paranetersT and (24"ro) ; the effusion rate tay. always be adjusted,
if necessary, by varying the orifice area a( while maintait•ing a fixed
geometry, ie.,, fixed values of T- and (L/re). This criterion for Optiin.
zation is directly applicable when any one of the following typical con-
dftion"s exist: (a) a very small. amount of sample is available; (b) intro-
duction of new sample into the apparatus requlres ýa comparatively long
down-time (as in mass spectrometry); (c) the materil2 itde-r study vaporlizes
incongruently and vaporization chiaracuerisi.c1s change with the compositrion
of the condensed phase; and (d) a low evap'oration coefficient i'posesC the
need to minimize the total flux from the cell, thereby minimizing the dis-
-lacereant of the actual pressure from the equilibrium value. Under each of
[I
these conditions it is clearly desirable to maximize the measured quantity
while simultaneously minimizing the total rate of effusion of sample from
the cell.
We shall now determine Lhad orifike gComleltry which maximizes each of
the following: (1) the near-axial flux density of the effusing molecular
stream; (2) the r~coil force generated on the effusion cell; and (5) the
force exerted on a targetU suspended in the effusing gas stream.
1. •timit.on of Moleculas Beam Intensity on an4d ZCr the Orifice
Axi s. In Part I1 and CFS S3 we obtainee cxprcz:hons for what we shall
here call the irremcntal transmission probability LWO, _.e., the
probability that a molecule which enters one end of the orifice will
exit from the opposite end into the incremental solid angle & located
at angle 0 from the orifice axis. With the solid angle ..w expressed in
spherical coordinates and the assumption of circular s}nmetry in the
distribution around the orifice axis, the equations may be wiitten as
dW0 = 2Q. Sine CosO dO; (a)
the n (n = 1,2,)) are complicated functions of the orifice parameters
L, (L/2o), and, depending on the range in which G lies, also of e.
The integral of dWe over 0 < I nr/2 is just the transmission probability
Y (designated W0 in Part I and CES to distinguish a calculation from
angular distribution considerations).
We now define the quantity Wy,
w • 0 0 2N% Sine Cose dO, (2)
which may be called the angular transmission probability, i.e., it is
the ratio of the number Ny of molecules which effuse per second into
a ccne which is coaxial with the orifice and has semi-apex angle h',
to the number No of molecules which enter the orifice through ro per
second.
The quantity to be maximized is the ratio of N- to the total
number NL of molecules which effuse from the orifice, i.e., NyiN1 a
and from equation (2),
w1/w - (2/W) 0 Q SinG Coso d. (5)
If we restrict Y to Range I1:' 4 (i.e., to 0 : Y < JTJ), %n beccnues 9a
12
and is independent of 0; equation (5) may be now integrated to obtain
W-Y/W = (Q2/w) Siw• 1 ; o 0 ITI. IQuite obviously, the fraction of effusing molecules which flow into
the cone defined by the orifice axis L. h aitile y can be varied by
varying y; the pertinent problem is to maximize that fraction for a
give" y determined by a particular experimental apparatus. Considering
then that Y is fixed, we re-write equation (4) as
Wy/(W Sin- Y) = (Q3/W), (o0 ! Y IT 1), (5)
and note that the quantity (QQ2 W) now to be maximized is a function
solely of orifice geometry.
A plot, prepared from data tabulated elsewhere1 4 , of (QVW) vs. T
for various (L_/_ro) is given in Figure 4. It is readily apparent that
a maximum exists at T Q 0' for each (.L/ro) and that the
maximum becomes sharper as (L/to) increases. We now note that under
the restriction 0 ! Y ý 1.T1, when T = 0', can have only the value
0°; hence, the points in Figure 4 for T = 0* apply only to the beam
intensity alS the orifice axis where Y = 0%
To determine whether W-Y/(W Sin2 •Y) is also a maximum at T = 0'
when )' has typical experimental values of 5' and 10', values of _WY
for cylindrical orifices (T = 0°) have been calculated. in effect by
integration of equation (2), but in fact from a closed-form solution1 4' 15.
The results, plotted as WT/(L_ Sin y), are in Figure 5 in which the
curves tor T - -i0' and for T Ž +10' duplicate those of Figure 4
since in these ranges (N • ITI).
It is evident in Figure 5 that for y = 5', _ Sin2 Y) is
larger for T - 00 than for any T Ž 50; similarly, for y = 10'
(_Y/W Sin2 Y) is larger for T = 0' than for any T - 100. It appears
to be true in general (although we have not made extensive computations'0confirm , "a s- e HT /_ N -- A -Itfy Y, Lj-2 A
L~lUJ LfIO VL a O U'lIu a 6 VCLt Y, ý= /. ýLL T j
is larger for T 00 than is (QVW) for any T Ž_! y.
For a short ori.fice (L_/ro < 2.0C and a given y, the two quantities
(_WE)T=- and (_W,/W Sin2 Y)T0 are tie same within 1i; for longer
orifices the difference is somewhat greater.
13
\U
i i I I I I I "* I *-******1
L/r'm ifIO0
10
4.0
6.0Q,W
3.0--4.0
i/- 0.2
-80 -60 -40 -20 0 20 40 60 80
T, deg
Figure 4. Collimating effect of various orifice geometries oneffusing molecules.
14|
U
'I I ! I I
VALUE Al T0"FOR y 5 0 '
50-- VALUE AT T=0 050-100° 10° FOR 1"&-00°
L-/rm:
.50-
100-40
80-
w
W Sin0T L/rm-
10060-80--
30 -8.0
60--
40-
,40-- 6
2 0 I4 o
-80 -60 -40 -20 0 20 40 60 80
"l)deg
Figure 5. Collimating effect of various orificu geometriesor effusing molecules.
15
Thcrc rcmai-ns the queLiuit vE ihe relative value of 04y/W Sin- y)
for T - 00 and for some T < y, e.g., for T - 0' and T 50 with Y - 10'.
For these cases (Wy/_ Sinp- y) is larger for the conical orifice
(T > O) and increases slowly with increasing T.
2. O•timization of the Effusive Recoil Force
In CES3 (equation 19) it was demoiustrated that the "incremental
recoil-force correction factor" dfo (designated dFU in CES), (i.e.,
the recoil force !• gerierat, A on a cell by effusion of molecules into
the incremental solid angle d at angle e from the orifice axis,
divided by the total recoil force 0i for the corresponding 4 ideal
orifice: dfr = do/oi) is related to dWo by
dfo = (3/2) CosG dWo
or, with equation (I),
df 0 = 3Qn SinG Cos 2 0 do. (6)
The (vttaWl,) xecoil force cr~reotion -factor";-9 is then given :by='n/2 3 bn
f = Y6=0 dfo = n~l f ar 3Qn Sine Cos 2 6 do; (7)
The integration limits (anbn) are discussed in CES and reference 14.
The factor f is, of course, the quantity tabulated in Table 1.
Optimum orifice geometry in this situation requires maximization
of recoil force for any given rate of effusion, which is equivalent
to maximizing the ratio (f/v). In Figure 6 this ratio is plotted vs.
_T for 'Vtiuus (".! l It is evident from Figures 2 and 3 that a maximum
must occur in a plot of (4/W) vs. T. However, it is rather surprising
that this maximum occurs at, or very near, T - +30' for a very wide
range of !1ro. It is quite clear from Figure 6 that the optimum
orifice for recoil force measurements would have a semi-apex angle T
of +30' (i.e., diverging) and would be as long as practicable (within
the range of 1/1o covered in Figure 6).
3. •£utuization of the Force Exerted on a Target in an Effusing
boLecular Beam: We BsbaL restrict our consideration to cases in
which the target in circular and coaxial with the orifice, and in
which the molecules striking the target either all condense or all
16
12 { " I I q i i I I |
L/rm 100- O
60/
1 20 -
20116
f
1.0
1 08-
9''f I
-60 -40 -20 0 20 40 60 80
T, de g
Figure 6. Optimization of orifice geometry for recoil force
Figure
I
rcvvaporize. The angle subtended at the •rifice by the target is
designatcd 2)'; ) then has the same meaning as in equation 2.
If all molecules condense on tle target, the ratio of the force
exerLed on the target to the force FO which would be exerted by
all wolecule. effusing from the corresponding4 ideal orifice, is given
by equation ('I) with integration over 0 < 0 • y:
S= FY/FO fy 3Qn SinQ Cos 2 0 do. (8)
It Y is restricted to 0 Y y • jT so that only (1 (- constant) is
required in the integration, equation (8) becomes
fy - Q1(1 - Cos-y).
We wish to maximize this force at a given flow rate and for a given Y;
therefore, we write
f" .
The quantity to be maximized is Q2H, as in subsection 1 above, and
the arguments given there apply.
If the molecules revaporize from the target, there will be
exerted on the targct an additional force which, with all extra-
orifice parameters fixed, will depend on the molecular flux onto the
target 3 . Maximization of this flux at a given total flux from the
orifice was the subject of aubsectlon 1 above; again, *•W is the
quantity to be maximized.
SECTION III
MEASLUREI.ý.NT OF ANGULARDISTRIBUTION OF MOLECULAR FLOW
THROUGH CONICAL ORIFICES
Of the various quantities which arc derived in the theoretical anal-
ysis 12. 1314 (see Section II) and which are amenable tv experimental study,
the most critical is the angular distribution of molecules effusing froman orifice, i.e., the variation with 0 of the number dN• of molecules which
pass per second from the orifice into the incremental solid angle dX located
at angle e from the orifice axis. Thie theoretical analysis predicts that
Sis proportional to Rn cos 0; _Q is the complicated function of orifice
parameters which arises from the non-ideality of the orifice (for the ideal
orifice, _Q, is always unity). Measurement of dN or an equivalent quantity
would provide experimental data which could be compared directly with the-
oretical values for 2n cos 0.
A. E)O?.RIMENTAL APPARATUS
To accomplish these measurements the apparatus described briefly in
this section (and in detail in Part 1Ii1) has been constructed. It is
designed to allow a study of the effusicn of a permanent gas (e.g., N2 ,
lie, CO,2 ) at any suitable pressure from any orifice with a geometry which
can be machined into a small circular plate. Permanent gases are used
as effusants So that the apparatus cani loe operated at room temperature;
concern that the reservoir-orifice systm may not be isothermal is there-
by minimized. For this advantage the ability to study the effusion process
as a function of temperature is sacrificed.
The angular distribution of effusinig molecules is determined by a
molecular beam method 2 2 incorporating a modulated beam technique2 -3 . The
reservoir from which the molecules effoac can be rotated on an axis which
4 -
passes through, and is parallel to, the outer face of the orifice under
study. Two stationary collimating orifices and the effusion orifice under
study define a molecular beam, the beam is modulated by a mechantcal chopper,
and its intenbity is determined by a neutral-beam detector.
The apparatus, a diagrammatic horizontal cross-section of which is
shoun in Figure 7(, consists of five principal components: (i) The Main
Vacum Chamber, which can be maintained at a pressure very luw wiLh respect
to the y-ressure in the gas reservoir, and wunich contains the rotating
effusing cell; (2) The Gas Reservoir, a large chamber from whiclh gas
flows to the effusion cell and in which the pressure can be kept constant;
(3) The Buffer Chamber, a small indCpe.ndenUtly-pumped chambLr which is
separated from the main chamber by a plate containing the first collimating
orifice, and which contains a chopper capable of interrupting the beam
about one hundred times per second; (4) The Detector Chamher, an independ-
ently-pumped volume which is separated from the buffer chamber by a plate
containing the second collimating orifice, and which contains an election-
impact molecular beam detector; (5) The Detector Electronics, which con-
sists of a power supply for the beam ionizer and a system to amplify,
measure, and record the ion current from the ion collector.
Each of these components was described in detail in Part I121; we shall
describe here only significant modifications.
1. Modulation of Mocular Beam: As experimentel techniques were
refined, it became apparent that the frequency of the mechanical
beam chopper (Figure 5, Part II) was not stable; the cause was
found to he an overloaded driving motor. When the motor was re-
placed by one with higher torque, a new motor mounting and chopper
housing (items 6 an,. 32, Figure 5, Part II) was fabricated from
stainless steel (rathel than brass). The new drive mechanism per-
formed satisfactorily at times, but at others exhibited erratic
chopping frequency. This difficulty arose from binding in the
bearings which support the choppel" shaft; the binding in turn
was caused by a slight misalignment of the new motor mount/chopper
housing when it was welded into the vacuum system.
At this point an electronically-driven tuning fork chopper
20
!
I
/-'8 |1'
17
16 L--JL----J Lm--- 7 ,-tJLw A/ /I1 13 1 r'04
2
16o~0 3 6 9 12 .. 1
SCALE -INCHES ____-
II
1. The Main Vacuum Chamber2. Ihc Gas Reservoir5. The Pluftcr ChamberL The DetecLor Chamber5. The Rotating Effusion cc1lU. Thi Beam Collimating Orifices
. ht: Beani Chopper8. The Bcam Ionizer9. Connectors for Tygon lubulation Whichl Carries Gas
from the Rcservoir to the Rotating Cell1'. Pumping ,1],'s Which Accommodate Liquid NitLrogen Traps11. Class Windows12. Rotary Vacuum Seal Through Wbich the Cell is Rotated15. Brass Bellows211. Clobe ValveI5' Copper pe , iameter1(t. Port for Attaching 5-Liter Stairtles Steel lank
17, Valve for Introducing Eifusant, GasJ,3. Connections to the Lqt.uibar Prtssure Mctei
Figu-re 7. Diagram of thc Lxpcrimental Appaiatus-
g2[
(type Wo) was obtained. atorianl frequency of the chopper is 320 ttz;
the chopping vanes attached to the tines are 10 mm high and have a
maximum aperture of 8-10 mm. An electrenic signal synchronized with
the mechanical oscillations of the tuning fork Is an inherent feature
of the driving circuit, and is readily available for use as reference
signal to the lock-in amplifier.
The new chopper was mounted in the BUFFER CHAMBER (hIgure 7),
buL directly o.n the flange which is welded onto the MAIN VACUUM CHAMBER
and which mates with the BUFFER CILAMBER flange. A new BUFFER CHAMBER
equipped with aluminum-foil-scaled flanges 21 was fabricated from
stainless steel.
No particular problems have been encountered wit00 the tunivg
fork chopper; it was used in obtai.ning essentially - the data
reported in Section B.
c2, Detector ChOsber: The Dctector Chamber descr'ibed in Part II
opercated sat.istactorily, except that the uliath•.e piessure was rel-
atively high ( L1&C torr). Conbequently, the life of the cathode
was seriously shortened and the emission current available from the
cathode was low. A new chamber was fabricated from stainless steel;
aluminum foil flanges"'' were used and all cla,;tomer 0-rings were
eliminated; a teore efficient liquid nitrogen trap ("Cryosorb") was
inserted betw.,een the 2" diffusion pump and the chamber.
"Aftrar a s!'art baec-ut at. I002--c' C, the Ce:w DatoCotor Cham1e-r
caes 5x1&" torr azd operates at 5xI0-7 torr vrith a bears, from
a 0.1 tc.irr source, entering the chamber.
The basic beam detector design and electronics have undergon:e
no significant change. In the improved vacuum system the beam
det!ct-.or hLas performed quite 6,atisfactorily; cathode life is iemark-
ebly ]engthened (ne qtiantitative data, as yet) and the emvission
current Is increased by a factor of live.
3. Valve,- The sensitivity of the bena detector cathode to oxygen
made it desirable to keep the detec(tor chamber evactuated coettlruouSl.y.
To achieve isolation of the Detcc; (.and But icr) Chamber whO le the
Main Chamber was opened, c.g., to change rcifices, a 0sii3ng, r (ri nmg
0')
seaIed valve wab instaliled in. the Main Chamber in such a way that
closu•e of the valve seaýled tl,h first collimating orifice from the
Main Cham-ber. The design of the valve was adapted from that of
Shecffieldut, It has worked satisfactorily.
)1. Alli•mont of Orifices, The apparatus was designed and constructed
to petro.it di.rect, visual alignment of the beam defining orifices,
if the Firad•ay ilon collector is removed from the beam detector
assembly. Aligrnent of the effusion orifice and the various colli-
matLing orifices is rather easilly accomplished by viewing with a low-
powered telescope along the beam axis. However, with only this check
on alignment it is possibhi foi the plane of a given orifice to be
tilted appreciably froln the i.,d perpendicularity to the beam
axis and the tilt. bc uuidot-e0table through the telescope.
Therefore, to ins11 parallelism. o. th; planes of all beam-
defining orifices while si. it! naeously establishing all orifices to
he coaxial, a second alig0ment -rocedure was adopted. A low-powered
telescope was fitted with a Gaussian eyepiece and the telescope
operated as an autocollimator: A lighr Source 1,n rhe side of the
eyepiece iluminmated a cress-haixr tlhe image of whitch was then pro-
jected through the telescope onto an optJcally flat (both sides)
front-surfaced mirror atteched fir•nly to the oiftce plate being
aligned; the oriiice plate was adjusted until the reflected image
of the cross-hair coincided i.n thC Cyepiece with the image of the
act-al ,'cto.ts-. hair; coincidence, o! t!he two imazgcs requires3 that the
mirror (ro-ifice plate) bc perpendicular to the liglhit (molecular)
beam.
I
-" a)
I
B. ELIER.IMENTAL RESULTS
Given the apparatus described abo.'e and the desired experimental
measitrement, i.e4, the atngular distribution of the effusing beam, experimental
procedure was rather straightforward. After a given orifice was inserted
into the Rotating Effusion Cell and aligned, the entire system was evacuated,
e.g., oveinight. At the beginning of a run the Gas Reservoir was ibolated
from the vacuum pumps (by closing valve i);. Figure 7) and filled with gas
(helium, nitrogen, nitrous oxide) to the desired pressure as measured by
a variable capacitance sensor (Equibar 120). Gases were obtained from high-
pressure cylinders; pressure in the reservoir was controlled WiL1I a variable-
leak valve (Type 9101-M) in the line between the usual cylinder regulator and
the reservoir. After steady state flow was established throughout the gas
flow system, the pressure in the reservoir remained surprisingly constant
(I to 2); a precision pressure regulator originally planned for insertion
in the gas line between the cylinder regulator and the variable-leak valve
was not required. Purity of the helium and nitrogen used was >99.5% and
of the N•O >98.0%.
While steady gas flew was being established, the electronic circuitry
was energized, adjustment of the lock-in amplifier checked, and, in partic-
ular, the filament current of the beam detector was adjusted to provide an
electron emission current of J.ClrA. The isolating valve (section III.A.5)
was then opened; the molecular beam could then pass to the detector and
measurements were begun.
With the gas piessure in the reservoir constant, beam intensities in
arbitrary units (i.e., the output from the lock-in amplifier) were recorded
with the Effusion Cell rotated to orientations varying by angular Increments
of 5' between 0' and + 9 0' and also between 0' and -90'. Typical concordance
between data for +e and for -0 is illustrated in Figure 8.
Angular distribution data have been obtained for five orifices with
thee .. t various Pressures; .... C.. . . .. 2 givcUs Ui d L of ti % lus
parameters.
In Figures 9-14 the results of various measurements are plotted as I
.ys. 0; _I8 is the relative molecular beam intensity normalized to 1.0 at
24
R U N I e'_'RUN 126]
ORIFICE NUMBER 1NIT ROGEN AT 266 dyn/crn2
00 0 0 E)O - 900
"9 CURVE SKETCHEDI90 THROUGH POINTS
80
70
I, 60-t40
.50
40-
30-
20 -
10
0 10 20 30 40 50 60 70 80 90
G, deg
Figure 8. Experimental angular distribution data (Normalized
beam intensity) for Nitrogen through orifice 1 (T - 25.65'
and (L/rrm) 4 4.010).
TABLE 2
ORIFICES, GASES, AND PRESSURESUSED IN
ANGULAR DISTRIBIUTION STUDIES
PRESSURE 0 oOrifice ,ASN,11ber T,deg L/rm r,,, Cal le NA NS0
= -2 -0
25.65 4.01o o.o46o 190., 70.0, 670., 200., 195., 69.0,45.0, 20.2 72.7, 4i.o, 45.0, 22.0,27.5, 20.2, 6.97.0
2 8.50 10.08 0.04.55 190., 71.0, 66o., 210., 207., 70.0,41.5, P1.0 70.0, 22,0, 41.5, 20.0,
7.0 6.23 -58.93 11.01 0.0292 230., 74.0, 700., 180., 690., 200.,
41.5 66.0, 14.o 69.o, 24.04 27.99 1.987 0.0951 -- 126., 12o.,
91.0, 75.0,4o.o, 24.0
5 0.0 2.2 39 0.0782 45.5, 53.0 70.0, 46.o,
25.5, 22.0,ti. '9
1.0 o He, 60.7d cm 2
-20.9 \ -/ He, 44.0d cm-,o . { 1(N ,) 11o d "-
N,93.3d cm
" 1. Q Cos f-L/r x 2.42
0.7 QNCOS L/r = 2.00
0 ?
0.
L~L 1 . l LI I L I0 10 20 30 40 50 60 70 80 90
8, Degrees
Figure 9. Experimental [1(e)] and Normalized Theoretical
[QnCose] Angular Distribution Data for T 0.00, L/rm = 2.h4.
27i
"o- .073 Torr
1.0 (8) a - .024 Torr
-A - .126 Torr
0.9 • QN Cos8 -
0.89
0,7
0.6-
C 0.5
0u 0.4
zuj 0.3-
020
0., -
01
01,I - 1 tI..
0 10 20 30 40 50 60 70 80 so8, DEGREES
Figure 10. Experimental [1(e)] and Nonnalized Theoretical
[QnCose] Angular Distribution Data for T = 30•, L/rm = 2.0,
with Nitrogen.
28
U
I I I I I
S- 8 27 dyn/cmn
2- 26 66 dyn/cm2100 1 ( G) 3- 55 33 dyn/cm -4- 93.46 dyn/crn 2
90 5- 2760 dyn/cm2
II OnCosoS80 -
F-4 \\ \
0 .70-
0 60 -\
0'50 \
S \ \\ 3
740 -\ 4
<' It
•'•-- "-•i
II0- 3C) -
10--
Figure Ii. Normalized Experimental [I(0)] and Theoretical
[QnCosO] Angular Distribution Data for Nitrous Oxide through
Orifice 2 (T - 8.5000. L/rm = ]O. O8).
29
T ------- I-
I ]- 18 67 dyn/cmr100 2- 8800 dyn/cm2
He) 3- 240.0 dyn/cmn2
9 , 4- 933.0 dyn/cm2
v \ \\\
80 Q~s
0 70 \\2 \ \
• o _ C/o
Li 50A A\\
N 50- 34 0 C
z \ \
S\\30 0\,
.20-
0 10 20 30 40 50 60 70 80 900, deg
Figure 12. Normalized Experimental [I(e)] and Theoretical
[QnCose] Angular Distribution Data for Nitrogen through
Orifice 3 (T - -58.930, L/rm - 11.01).
30
I1- 32 00 dynicrn 2
2- 92 00 dyn/cmL
100 N\3- 266 0 dyn/cm 24- 9200 dyn/cm
2
\ ,\.90 X,,\QnCoOS
\,,,.80\ ' \,o 70 \ \\",
\\\ \ \\\\\ \ \,
.60 ¶
a)5.6 \ \\
C.)
Ld 3.4 N3,0- \ \
N\ \j\A\
cr \ \",.\
0 30 2 \ ,\\.
~~A\ \ ,\\
.20- \ X,
\~\
0 10 20 30 40 50 60 70 00 90
0,deg
Figure 13. Normalizet. Experimental [I(e)] and Thlorctical
[QnCOe] Angular Distribution Data for Nitrous Oxide through
Orifice 3 (T = -56.95°, L/rm = lI.CI).
51
1 - 9 33 dyn/crn 2
100 2- 26 93 dyn/crn 2
3- 36 6G dyn/cr2
1(0) 4- 54 66 dyn/cm290C - 5- 96 93dyn/cm2
/ 6- 26G 6 dyn/cm2
I-- 7- 8933 dyn/cn 2
so - ~ 7-/-
Q3 70 ,
<:0D 6O
t-O5 0ncos\
C70
0 50
140 "'
O& ,
7
o) 30
0 I1 20 30 40 50 60 70 80 90 i
202E),deg
Figurc 14. Norn-alized Experimental [I(e)] an, Theoretical
[QCos8j Angular Distribution Data for Nitrogen through
orifice 1 (Tf = 25.C5*, L/rm .4cl)
0=00, i.e., I -M /_+o, with-j0 representing the measured ion current at o.
In Figures 9 and 10 the experimental points are shown, and the lines
are drawn through theoretical values. In Figuires 11-14, the dashed lines
represent smoothed curves drawn through the experimental points for both
positive and negative angles; the scatter of points about the smoothed curve
in Figure 8 is typical, i.e., not the minimal. The solid line in each
figure is drawn through theoretical values which are discussed in the follow-
ing section.
C. DISCUSSION
1. Remarks on T : Implicitly in Part 112 and CES1'3 and explicitly
in Reference 14 the theoretical angular distribution is expressed as
dNo(L) - 2qto Qn Sine CosO do; (9)
dN0(L) is the number of molecules which effuse per second into the
solid angle al1 Sine do, po is the molecular flux incident on the
entrance of the orifice, and Qn (n - 1,2,5) is the complicated function
of T, Lro, and 0 which describes deviations from the "ideal" cosine
law distribution. We now define
QV Q Qn/n (v V n - 1 2, 3), (i0)
insert ro explicitly (ro and L are normalized to ro - I in the theo-
retical analysisl2" 4 ), and write in terms of a generalized incre-
mental solid angle dx:
dNo(L) - 1onro2 Q, Cosa (2n Sine dO) (n:)dNo(L) = •olnr 0
2 Q, CosO dw.
The transmission probability W may be expressed as (number of mo!eculee
effus,,• from orifice)/( numbe enterin, g nrifce), r with eqa, n (1))
W (14/onrc 2 ) f PoTro-2 Q% Cos du
-" Cos coe (b.
- �/�Q. Coso (2v Sine do). (c)
Equation (12c) may be compared with equation (1), if equation (10) is
noted.
5.5
We assume that the aperture of the mokLcula." beam detector
subtends Pt the effusion e-ifLce 6 solid angle (f2d) sufficiently small
that Q CUSE1 may be considered constant over d, For two anglets G
and 8' we may new write
N•(L) ' n y"ro2 Qv' Coso' ;d (a)
and (1.)
N-f(L) , o % coseo Cjd. vCb)
The ratic of equation (15b) to (13a) is, if 0' is ta!:en saqt
Equation (i14) gives the th'eojetical value of the ratto o2 (number
enteriilg dptec or at O) to (number e;itering rt e - C').
"Ihe basic assumption concerning the operatLon of the bezia
detector is that the mjasured ion current I_+ is proportional to the
number of neutral melecule- entering the detector, or
- 'e (15)
We now define the symbol * to mean 'is (theoretically) predicted to
be equal to", an6 combine equations (]4) and (15) to obtoin
QV ~Cosa (16)
For any orifice, at 0 = 0' equation (20 becomes
la= 1,+ = 1 t (Q)o • 1 - 1; (r7)
hence, both the experimental and the theoretical, resul•s are aelf-
normalizinr, to unity at 8 - 0°. Equation (16) Ia the baails for the
form of the graphs in subsect on B.
As indicated in equation (1y), the plotting method suggested by
equation (16) and used in Figures 8-11', forces sgreement between
experimental and theoretical results at 0 = 0'. Furthermore, the
nature cf the experiment essentially forces agreement at 8 = 90*.
Therefore, whatever the actual nature of the discrepancy between
.5'4
experimental and theoretical results, in plots ot 1I anfL_- Co°o/QQv)o
3 E- 0 the apparent discepancy near 0'O0 is small and any real
discrepancies are forced to appear in midranger of 0 and are therefore
overemphasized. To circumveut this difficulty Phipps and Adamsfl'3 have
introduced, and Wahlbeck and Waug'"' have also used, a probability
density junction Po which we now conaider.
The experimental probability density function px is defined' "a by
PX = . +'1'!/-2 1 + (2, Sine do). (].8)
Obviously, I f normalized ion currents (cf. equation 15) are substituted
for T P6 ia unchanged:
/ I C./ I (2nSio dO) (19)
We alec note that st.uce I_ 3.0-90
l/e'~I (2, Sine dO) (20)
and
xX
It is apparent from equation (21) that 9 plot of Px vs. e will
differ from a plot of I .s. X enl% y by the factor pX. For a given setpx
of LiO, 91 data fo is obviously fixed, but there i no requiremenL on
the constancy of PX from run to run; hence, Px is not self-normalized
and can reflect di:--repanciee between experiment6l and theoretical
data at e - 0° as well as at other valuez of L.
The interpretation of--9 as a probabilicy density funct-jon fol.iows
immediately from its definition (equation (18) or (19)): P" dw Is
the probabqlity that an effusiE molecule will travspeicc. duo at 6,
or stated differently, _g is the fraction (of efu.ýsirt molecules)
wich Hlow, per &teradian, at e. With this_ interpretation of _P-nd With ...... '- • . ... . .•aand.. wtl%,i aantl an l3Ye5Si [I)t fo•- tile cureapording theoretical
quantity, t may bc obtalned:
t= dN,) (L) ,. !L1°r°•2 Qý, Coso d
dw¾oo' W
or,
tQP (Q Cos)/W (22)
351
I
The quanitity Pt also is not self-normalized, but may be converted
to a self --nor•nalized form by multiplying by ,/(0 •" the result ia
then to be compared with equation (16):
W Pt/(Qv)o - (QCoso)/(Qv)o t Io (1
FroM equetions (23) and (21) one ob:ains
x (Q)o/W. (24)
From the above obteorvaLions we draw the following conclusions:
(i) A plot of P vs. e revcals no information not alreadyx
pro:ided by a ploL of .9 vs. 0 and by the value of P0
(cEf. eqatlo 21)
(2) If one. attempts to nornalize V- via equation (25) and compare
the i-e.lilta with ]-G (e.g., 81 1(I - P'XW/(Q,)o), one obtains
nothing new, since
= ~V(.o ~~i-PW/(, o (25)
and the bracketed port:ion of the equation is a constant
for a given run. If, en the other hand, one attempts
to compare I directly with _•(t )O (equation 2)) by
defining
0/ [Q6 -QV)o],
one finds that
=[I(Q )/WPt] - 1, (26)
and, by virtue of equations (17) and (22), 8,2 is inhcrently
nornalized to zero at 0 - 0'. Furthermore, a plot of 82 '_.
i is, except fur a shift in the zero point of the ordinatex
caused by non-identity of P0 and (,)o/W (cf. equation 24),
ideiica" to the more useful fovin now to be descrj!bed.
(5) The moat informative scheme for comparing eXperTimental and
theorecical results to that obtained by defining, and plot-
ting Is. C, the quantitiee A and f(A)
x t
f(A) - A/P ( - (-
A plot of f(A) vs. 0 provides a direct, non-normalized
indication of the discrepancy betweenL experimental and
theoretical results, expressed as a fraction of the theo-
retica value. The only restriction on the value of f(A)t xis that -oo Z 0 and Pso (i.e., I~o) is adjusted to zero
by subtracting background ion current; therefore, A 0.
The one difficulty presented by the use of f(A) is thatnx tat E > --70' P0 and P may be rather small (e.g. 0.02 of
the value at o = 00); the difference A may be small com-
pared to experimenta( egror in ob0aini2g 1., but b( A) canbe quite large (e.g., 0.20 - O.LO) because PG is small.
In other words, the fractional errors are magnified as Pt
approaches zero. for this reason, one might prefer to
plot t va, 0; both types of plots are illustrated subse-
quenL nly.
2. Exiicrimenta] Results: Figures 15-20 present various portions ofx tou- results in the form: (P-/Po) - I vs. O, i.e. f(A) vs. e, except
for Figure 16 which is a replot of Figure 15 in the form (P -) vs.
f, i.e., A vs. e- Comparison of Figures 15 and lb illustrate the
point made in the preceding suction: the large excursions of f(A)
at 0 > 70' do not realistically reflect the dEactepancy between P x
and P0. At 0 < 60' plots of A vs. ( and of f(A) y.-. 8 are very simildr
except for scaling.
Figure 21 is a replot in the form [(1 0 (Q )o/Wpt) - 1] i. e,iLe. 6 .. e, (cf. equation 26) of Run 125 in Figure 15, and is
in(luded to illustrate two points: (1) a plot of 62 vs. e is nor-
malized to zeto at 0 = 0', and (2) except for point (1) and scaling,the shape oL the curve is the saame as that of f(A) va. e in Figure 15.
B~efore we examirte- the V'riou's curves in dec1 it =y be useful
to note again the geowetry cf the corilcal orifice and the two important
angles T and U*; thuse aie illubtrated in Figure 22 for the diverg-
in, orifice (T > 0), For G 1 e : I, tke detector may receive molecules
, froed 1lU of tihe ciiv.lar e:ttrance (a the oifLice and frow all
elements of the onificL wall; this coý iespuud~li• to Renge 1 and n - 1
/
06 I,
! 20 SERIES
( L/ro) = 401
0 4
N lROI "OE N
III
02-
p.-- I,/;.I,I,
I,7
I / ',t"- _ •
-04 - R N i,
0 12 66
A 127 8933•0
- -- --- I ----------L
5 I . 5
0 10 20 30 40 50 60 70 80 go
E), degFo
02ue]. f()pot o 2 eis
0a
4
120 SERIES030 (L/ro) --- 4 D6
8 NITROGEN
020 0.5
010- -04
0p0.
7 - ~
-010- O02
PRESSURE,
"0 20- RUN drycm2 01o 121 9.33
O 123 36660 12 I5 9693L 127 893 30
- I0 10 20 30 40 50 60 70 80 900, deg
Figure 16. A plots for 120 series.
I ~ I
230 SERIES
04 -o (L/ro) z 10.08
NITROUS OXIDE
02 ' 7'
i I -,
-02
'd PRESSURE,RUN dyn/cm2 \•
o o 231 8 27 I- 0 232 2666
-06 o o 233 55.33 ."" " 234 9346
v 235 276.00S I I I I 1 I. .~~ j
0 10 20 30 40 50 60 70 80 900, degi
Figure 17. f(A) plots for 23C series.
4o
320 SERIES0.6- (L/rm) = I 101
NI TROGEN
04 4OC
0
! i)
0~0
0.- 00
-021•
PRE55URE.-04 -RUN dyn/cm2 \'
o 321 1867 3o 322 88 Ou
A53~ 24000C' 324 93330 \2
-C I I I0 10 20 30 40 50 60 70 80 90
Odeg
Figure. 18. f(L) plots !or 5ýC series.
24 1
"0 G
o C i i I =- r
4 330 SLRIES-(L/rr) 1 10NITROUS OXIDE
0-1-
0~
-0Th
3PRESSURE,
-04- RUN dyn/crn 2 ,
o 331 3200 0o
[] 332 9200 Ln f
6 333 26660 It
) 334 92000 F- cD
0 10 20 30 40 50 60 70 80 90O, leg
Figure 19. f(L) plots for 33C series.
I I I I I I I I I I I II I I2
420 SE.FILS011- (L /ro) = 1,"99
02
0~0
-02A
CC
V L(0
*CF
PRES SURE,RUIN d yn /cm1
2
0 42 1 3200-06- -1 23 97 33
on .26 168 00
0 ic, 20 '30 40 -50 60 70 80 900, deg
Figuc 2.f(L) ploLth for .Xseiic3.
i / -
04 /
02-
O2
"--. RUN 125
0
-02
0 i0 20 30 40 50 60 70 80 900, Oeg
Figure 21. &: plct (cf. equation :'C,) for Run 125. Comparewith Run 125 in Figure 15.
4414
DEl [EC ORA PERT UR Elý
GASFLOW Lz-
j E): T
II ;
/x-
E) ?I J ~ * *// / // // JI
2 1, /"
"I,' //! / "
Figur /-I T,, .,_/ A/ a
8 C i Orifice//\
, \
/ G•_T \
Figure ,;2. The Criti cal Angles and thc Angular Ranges for
a Conical Orifice
(cf. equation 9 and 10). For T <. a < e0, the detector may receive
molecules directly from elements of the orifice wall over the entire
length of the orifice and frow par of the entrance to the orifice,
but a portion of the wall, and of the entrance, are shielded from
the detector by the outer rim of the orifice; this case corresponds
to Range II and n - 2. Finally, for o0 g 0 o T/2 the detector
receives molecules 2oly from 2_ of the orifice wall; molecules which
traverse the orifice entrance cannot proceed directly to the detector;
this is Range II113, and n - 3. For the converging orifice (T < 0),
the above statoments are also valid (i) if T is replaced by ITI, and
(2) if the description of Range I is modified to read: ,,receive
molecules directly only from - por~tLon of the ocifice entrance".
The significance of T and 0* with regard to angular distribution
m iay be summarized: For 0 < -, II elements of the orifice wall
(_T > 0), o: no elements (• < 0), contribute to the flux at the detector
aperture. As .jfp•rLeta in the range ITs 0 C * a decrtai
portion of both the orifice entrance and the orifice wall contribute
to the detected flux. For 0 > e*, the orifice entrance makes no
contribution, all molecules reach the detector from a portion of the
orifice walls, and this portion decreases to 0 as e -4 •/2.
We shall now examine the features of the curves in Figures 15-20,
beginning with Figures 20 and 15. The agreement between experimental
and theoretical values for P0 for the 420 series (Figure 20) is very
good except in the range 40' • 8 • 600 where there is a maximum inX tI
each curve. Similarly, in the 120 series the discrepancy t, P9 - P0
for the lower pressure runs (121. and 12 )) is zero within 2 to 4%,
i.e., very nearly within experimental error, for.q <,T; however,
there is again a maximum in the curves at @ i- 40'. At higher pressures
(Runs 125 and 127) f(A) is -0.05 to -0.10 at 0 <T, the maximum in
each curve is high' e and ib ishifted to larger 9. We now note the
correlation, in Figures 15, 16, and 20, of the maxima with the angles
T and 0*, and that the magnitude of both the maxima and the discrep-
ancy f(6) increases with increasing pressure and with increasing
46 1
Support L foT .is last gencial i zat •ln wati soight in Wang's data
for t11e effusion of cesium chloride !hiough 1 cylindiical (T ',. (. )
copper sld nickel orifices; data required for the t vs. I plots in
Figures 2" and 2'4 have been calculatd• by us fr +/1 + del~~~ .... /, u0fou ;G] dr-ta
tabulated by Wang-7. Maxima in the various curves at 10-15* and min nima
at (00-70' are immediately obvious (but note that the L scale is expanded
by a factor of five compared with Figures 15-21). Thc discrepancy A
does indeed appear to dec-ecase with increasing pressure, but this asscrtion
must be qualificd: Run 75 in Figure 2ji i- "out of order" for no apparent
reason; in Figure 25 Runs 21 and 2u were made with one experimental
configuration, Runs 2i', 28, and 55, with another; within each group
A decreases with increasing pressure; the reason for the discrepancy
between the groups is not apparent. The shift of the maximum to higher
0 with increasing pressure is qujite evident tn Run 91, Figure 214, hut is
not clearly exhibited in Figure 23, perhaps because data were not taken
at sufficiently high pressures
It might appear that Wang's data, then,provide support for, if not
confirnration of, our generalization about the variatiou of A with (vL/r)
and with pressure. However, we were surprised that for low pressures
the maxima in Figures 25 and 2b occur in the same angular range (10-151)
despite the difference in (L_'r) for the two orifices. Upon investigation,
we find that in Wang's apparatus7? the angle from the orifice to the annulus
which surrounds the baffle plate in the front oven is in the range 10 to
15'. It is therefore not clear whether the maxima in the curves from
Wang's data arise from gas ilow-orifice phenomena or from the baffle-
ennulus acting as a (reatively) concentrated source of molecules.
An obviouti question, then, is whether the maxima in Figures 15, 36
and 20 arise from a cell effect rather than an orifice effect. The
cylindrical iuterior ot our simulated Knudsen Cell is 1.00" in diameter
and 1.10" high; the bottom of the cell is completely open to the flow
nf g- f'nm I-t 14,i The int-lor ,.1 of the cell is macro-
scopically smooth. Hence, the only discontinuity in the emitting surface"seen" by the detector through the effusion orifice is at the "bottom
corner" of the cell; the angle front the orifice to the "boLtom corner"
47
l~14 I I T I
VNt, NG'S COPPER ORIWICEI2- (L/io) 0934
CESiUM CILORIDL
!0
08
•.7
00
04
--T- 0 00 PRESSURE,RUN dyin/c 2 -
V 21 1 30S27 27 0 '1
o 28 1330 *CD
0 26 2670o 35 3330
IOL I ,
0 10 20 30 40 50 60 70 80 90
Figure 2j. A plot for K. C. Wang's angular distribution data'-,
for otifice with T = o.0° and L/ro = 0.954.
48
141 -r- i r r-- i 1
WANG'S NICKEL. ORIFICE
12I\ ( L/ro) : 2 59
59 CESIUM CHLORiDE
00
i
7\
Oc',
O04 1l•i/ • / "
02'"I~ /
g° /0
02-
- 0
0
z HPRESSURE,
O RUN dyn/cm 2
-08V 75 2 70o 67 400
-10 C3 59 54 7r-
0 7!1 1 ,O0n I1I 409 30
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
I0 20 30 '10 50 60 70 80 90
l'ig ic g I .C A plot fto K. C, Wag'Igis angul ai distlil ution data7
fo ] or i ice With T 1 0.Q0 a;id ./ ) - ,
is -25°. For 0 greater than "--•Q molecules can travel directly irtxu
-he cell wall through the orifice (it 0* > 25') to the detector; for 0
less than •25° the detector cannot "see" the cell wall and receives no
molecules directly from it. Wc now examine Figures 29 and 2D for evidence
ci a cell effect an find that in Figure 16 V is so near 25' that one
cannot hope to distinguish cell effects from orifice effects; in Figure
20 the eituatiori is similar. However, the data for Figures 15 and 19
were obtained for a converging orifice with T - -58.9' ; in this case the
orifice presents no obstruction to molecules traveling along a 2 • 250
trajectory from the region of the surface discontinuity to the detector.
Figures 17 and 18 provide no evidence (i.e.,, no maxima no minima, no
breaks in thle curves) for a cell effect at or near 0 - 25', but there
is a definite change (more pronounced at higher pressures) in the curvature
of the various curves in the narrow range ITI a * 0* . We therefore
conclude that the deviations from theoretical behavior, exhibited by the
curves in the various figures, are orifice effects.
We return our attention to Fiires 16 and 20 and consider the two
mechanisms frequently cited as causes of deviation from theoretical
behavior: specular reflection from the ovifice wall and surface diffusion
along the orifice wall 9 . One would expect any enhancement of intensity
by specular reflection to uccur at angles 0 ý_T. Enhancement at angles
O > T requires specular reflection through angles greater than 2T, and
the maxima of Figures 16 and 20 would require 2L~f~etkg• enhancement
by specular reflection through angles > 50* with no concurrent depletion
at smaller angles (jr. especially Figure 20). Available experimental
dataeS on angular distribution of molecules reflected from surfaces do
not support these rather stringent requirements.
If surface diffusion occurs along the orifice wall, the concentration
of molecules resident within any incremental area on the wall is expected
to be greater than if surface diffusion did not occur; consequently, the
flux from the orifice wall to the detector Is expected to be greater.
Therefore, one might expect any discrepancy between theoretical and
experimental data for angular distribuLions, which results from surface
diffusion, to be most prominent in the range(s) of 0 in which the flux
50
from the orifice wall rc6a makt.5 thu MaXilifll tCl8LivL cou1Libuton Lo the
total flux into the detector.
The area cA the projection of the orifice wall onto a pla'nc per-
pendicular to the orifice-detector line defined by o may be calculated
from the following equations (sec Figure 1 tor Symbols):
AL - TrL Cos 0
A0 "rol, Cos C (28)
AE A(DIo; t 1 ) Cos 0
[A(Do; DL) is tihe area of overlap of the circlC T1 -1 , with the pro-
jection at anglc 0 onto the plane of that circle of the circle J~rL;
derivation of this quantity has been described in detaill32, 3•4 .4
1i f ~ Aw - A L
I< 0 <i 0l, Aw = AL - AE
if e a Iii nd T > 0. 0, A, - Al, -A
If 0 a•i'- aid T < 0.0, Aw = 0.0
The resulls of these c ltions for the eril-ces ior whi --2.
distribution dat.a were obtained are shown iln Figures 25-26 wherein,
for convenience in plotting, the areas have been nornalized to AL( Oo.O)-n.O.
In the ranges 0 > 0•* and 0 : T, the variation of A,, with 0 is given by
Cos - lowever, in the range T o - ,Aw exhibits a rather different
behavior which produces a miniiinim (or virtual mainituim) int Aw at 0 -
and a ,naxinli.m (or virtua! .ii4xi,,u,,) at "
As 0 increases over tl.c range 'i 0 0 4z for a diveigi ng (T > O.O)
orifice, the entrance of the oriflice is eclipsed by thle outer rim of thle
exit, and therefore the fLtLoIsl contr-ibuti on from the wall to the
total flux to the detector increasos to uniLy at 0 C,4 1 - In the range
o g 0t) over wlichi the fractional coil. cibutlon from t he wall is uni ty,
the projected wall a "seen' by the d2tector is a maximum at 0 U"
Therefore, if there were an increase ii, the flux Irom the oritice wall
above that predicted by our extension of Clausing's Model, one should
expect to see maxitua in Z vs. 0 plots t' or near e =
This is precisely what Is observed in Figures 15, it.), JY, and 20, and
in lieu of any acceptable Llternative. we tentatively conclude that the
51
U
00 -- " -- , , , I90o
80
70o-
LLJ 5Q -
10 h.30-
20- ej
I0 __
0 . I j de g. •_-•
0 10 20 Y) 40 50 60 70 80 90E), d:eg
Figure 25. Normalized projected area of inside wall of Orifice 1.
52I
100 I I
90-
80
7J 0
GO
LLJ bo
ODD
30 Ar-
20-
I0
0 1 ____.___._____. __,.___.__._____._.0 10 2.0 30 40 50 60 70 80 90
0, deg
t Figur-e '(6. Normalized projected arca ol inside wall of Oiit•'t,
I
1 0 0 - ---- -
70
60 -
750
,40- °
00 o
,20
0 °0 10 20 30 40 50 60 70 80 90
e,deg •
Figure 2•(. Normalized projected area of inside wall of Orifice 2.
54+
,I
I00 - -T !
90
UOC
60
C a CO30 Goo
20
30
20
0 10 20 30 140 50 60 70 80 90C), deg
Figure 28. Normalized projcctcd area of inside wall of Orifice .
!I
,855i
I
increased flux from the orifice wall results from surface diffusion.
We have not yet made a more detailed analysib of a model including Surface
diffuaLoa; Ruth and lhrth 2 E0 have reported the corresponding tinalysis for
cylindrical orifices.
Other features of the f(L) vs. 0 plots, e.g. variations with pressure
and nature of the gas, are being analyzed and will be reported in the opeu
literature.
56
fU
SECTION IV
TIE MUITICELL TECHNIQUE FOR EXPERIMENTALDETERMINATION OF TRANSMISSION PROBABILITIES FOR
MOLECULAR FL OW TIIROUClH CONICAL ORIFICES
The Knudsen effusion techniquei' 3 has been widely used to obtain
vapor pressure data in the range 1C-- 1Q" atm. The technique has several
variants 3 , but basically the experimentally measured variable is the
isothermal rate of flew of vapor through an orifice, e.g., I nan in diameter
and 0. 5 mm long. The equation which relates the pressure of the gas to
the flux through the orifice contains a factot which is often called the
Clausing factor, or more properly, the transmission probability, of the
orifice, i.e., the probability that a molecule which has entered one end
of the orifice will exit from the opposite end. Numerical values for the
transmission probability are available from theoretical analysess-B 12-- 14
of molecular flow of gas through orifices with various geometries. Although
these theoretical transmission probabilities are widely2- 3 used to correct
the observed rate of flow for the effect of finite o-riflce icigtih, reliable
experimental data with which to compare the theoretical transmission prob-
abilities are scarce, 3 1 ,33.
The Multicell technique, which we describe below, provides a simple
method for obtaining experimental values for transmission probabilities
and also offers a means for critically testing various assumptions which
have been made9430 concerning molecular flow within a Knudsen cell.
Application of the technique to the determination of experimental transmission
probabilities is discussed in this section.
A. THE MULTICELL APPROACH
In a Multicell effusion experiment a number (e.g., 4 to ]0) of ef-
fusion cells2, 3 which are as nearly identical as possible except in the
57
4|
variable under study are subjected simultancously to a given environment
for a given time. Comparison of results (e.g., weight lost from each cell)
for the Pvveral cells in a giv• n experiment permits isolation of the
effect of the variable of interest since all other variables are constant,
or vary simultaneously and identically, for all cells.
In the applicaLion of this approach to the determination of experimental
transmuission probabilities, Knudsen cells are fabricated to be essentially
identical except that each orifice is different from the others; each cell
is loaded with an equal amount of a given sample (e.g., cedmium or mercury),
and then heated simult.neously in vacuum for a givcn time - and at an
appropriate Lemperatur2 e which are identical for all cells. The weight
lost from each cell during the experiment is determined by weighing before
and after the heating period. The weight di lost from the ith cell in
time dt is given by'3 7
dg, ý aiKI[P(M/2TTRO) 2dt] (30)
in which P is the pressure in the cell at temperature 0, M is the molecular
weight of the effusing molecule-, Ki is the experimental transmission
probability and ai the cross-sectional area for the ith orifice. For
the experimental conditions described above, P, 0, and M are, at each
instant, the same for all m cells; therefore, the integral of the bracketed
portion of equation (30) over the interval 0 to T is the same for each cell.
Hence, if .gi is the total weight lost in time T, we write
.rt=o,t=T dgi -gi = aK I
with I representing the integral which is constant for the several cells
in a given run, but which varies from run to run. Since the various 8i
and gi are directly measurable quantities, it is convenient to define
-- i/Ai and then to write
Ji = K- (31)
Clausing's theory predicts5- P 13 that, for conical (-!i/2 < T < rr/2)
and cylindrical (I .- ) orifices, the variation in a given run of the values
Ji for the various cells depends only on the ratio if length L to diameter
D for the orifices and on T the semi-apex ongle of the cone of which the
58
p
orifice is a truncated section. ieiz-fore, if the values if are plotted
axainst some suitable function of (IQ,1) and T, designated by (I.'PDi), andextrapolated to o0, T) one 1ay write; fI111 Eqti.atin (51)ý Jo I 1 i hAch
the subscript designates the condition L ,n = ". h'tL (L,' = O defines5-tý, 13
tiL ,,ideal,, orifice for which tie transmission probtlilltt, X. is unity
1vgardless of the value of T (whichl in fact it mesninglcss when LiD = 0);
therefore,
Hence, the intercoPt at FN 0 ) is just the wcight loss per e'it area Ji
to be ex1pected, under the conditions of the run, frot, an idevl orifice.
With this cxperiment.,] value for i, the experimental transmiss 4 on probability
for each of tle cells is calculated from Equations (5i) and (52):
K, = ai/ao. (33)
Tie i11ert suitable choice for (i•, ),T) appears to be 14, tile theoretical
tra:,smission probability S-u, 13 calculated from the orifice dimensions. This
choice results in a conveniently bounded plot with abscissa values ranging
from zero to one; ':.ie e important, if the theoretical transmission probabilities
predicted by Clausing's theory5 and its extensions(3 '0 4 , 13 are correct, a I-lot
of Ji versus F(LL/D,T)j!. -= WT should give a straight line which may be precisely
extrapolated to ji = 1 (i.e., to L/D a ) to obtain the ir-ercept 30. The
slope of the straight line resulting frov, such a plot should also equal Jo.
B. EXPERIMENTAL RESULTS AND DISCUSSION
The results reported here were obtained with a typical Multicell
configuration: various sets of eight nickel-plated, steel Knudsen cells,
machined to be "identical" except ior the lengths of the cylindrical orifices
and/or the semi-apex angles T of the conical orifices, were placed into
closely-fitting, symntcLrically-arranged cavities in an aluminum (or copper)
block which, during a run, sat inside a s,:ainless steel vacuum chamber.
During the heating period, the entire vactuum chamber and several inches of
connecting stainless steel vacuum line were inside a circulating-hot-air,
constant-temperature oven. The cells were charged with equal amounts of
high-purity cadmium metal.
59
The dimcnzions of the orifices and the detailed results of the various
runs are available elsewhere 3 0 ,40; the results of a typical run are presented
in Tablc 5 and in Figure 29 in which the lenst-squares straight line (Jic -
mW1 4- b) for the dat.a points ie also given.
For each run, the expeiimental transmission probabilities were calculated
i1 two ways: (2) the :,rdinate value of each experimental point (i.e., each
i was divided by j0 (cf. equation 33) to obtain the values Kip; (2) the
coriespondin ordinate value (Jic) on the least-squares line was divided by
Jo to obtain the values Ki C. In Table 5, the transmission probabilities cal-
culated in the two ways from the data of Run 7 are compared with the theoreticalA
transmission probabilities.
For each orifice the transmission probability calculated from the results
of several runs will be more reliable than a value obtained from any single,
arbitrarily-chosen run. "Average" transmission probabilities Ki for the
various sets of orifices, and the transmissioa probabilities Ki calculazed Ifrom the individual runs are given in Tables 4-6. Two values are given for
each orifice: the upper of each pair of numbers is the transmission probability
calculated from the experime ntal poin', (2f. Kip, Table .5), and the lower
number is the value calculated from the least-squares line for the points of
a given run (cf. Ki , Table 3).
As shown in the last culumn of Tables 4- 6 , the discrepancies between the
experimental values for transmission probabilities (foi cadmium vapor passing
through nickel-plated, cylindrical orifices) and the theoretical values lie
in the range ±-% (with two notable exceptions in Table 6, for which we have
no explanation), if the experitental points are considered. If th3 results
from the least-squares lines are used, the discrepancies are appreciably
smaller. Similar agreement for a very limited number of "short" orifices
has been obtained by McKinley and Vance 3 3 and by Carlson, Gilles and Thorn37.
In 6cicral, the experimental values K- are less than the theoretical
lL, Lnd thlere is a sligit trend for the discrepancy to become larger as
increases. These observations are exactly opposite to what would be expected
if surface diffusion8'38 through the orifice contributed significantly to the
total efflux.
The agreement between theoretical predictions and the experimental
results reported here (especially Table 4) is appreciably better than the
60 o ,I
i
TABLE 3
COMPARl',ON OF EXTERIMENIAL AND THEORETICAL!'l-f-'MISSION PROBABILITIES: RUN 7
Orifice, 300(0K-14] 'oo(ycw).L/D Wi gng .' y W
0.195 0.83Y 42.36 0.843 +0.7 0.831 -0.8
0.290 0.77W 38.16 0.759 -2.3 -.767 -1.3
0.414 0.711 35.62 0.709 -0.3 0.69) -1.8
o.048 o.695 32.94 0.656 -6.o 0.682 -1.8
0.5521 o. 664 33.83 o.673 +1.4 0. 649 -2.2
0.522 0.663 31.91 0.635 o.649 -?..2
0.667 0.609 28.85 0. 574 -6.1 0.592 -2.9
0.8oo 0.567 28.55 o.568 +0.2 0.549 -3.4
61
, .
I
500- J0 50 26~
450-
I
400
S, -mg
350
300- , -
060 0.70 080 1 or)
W1
Figure 29. Weight loss through orifices with various (L/D)ratios vs. theoretical transri.ssion probability.
62
II
TABLE I
EXPERII.INTAL TRANSMISSIONI PROBABILiTIESFOR) FICPT CA!,DP ' ORIFICES (SET I)
Orifice, K I_(_o -W)
L/D W Run 4 Run 5 Run 6 Run 7 K W
0.195 0.835 0.85519 o.B,512 0.8H50 0.814:29 0. N46 +1. I0 . 629(b 0,8392 0.-3 78 0.8H305 .83) -0.4
0.290 0.7K( --- 0.7207 0.75h6 0.7"75 0.745 -4.]--- 0.7791 0.7772 0.707 0.'7(5 -0.5
C. 4.34 0.711 0.69)2a 0.7395 0.6888 C 7(k 0.700 -0.40 .6-) 97 5 b 0.7±41 0.71'16 0. 0.706 -0.7
0. 448 0.695 0. 6 651)a o. 0. 991 0-(555 0."670.610 b 0.698 0.C60)o 0, 6)24 0.690 -0.
0. 521 o. 664 O. 0- 610 .o74 o 5 0. 6905 0. U' _ 2 . 689 +3.6o.6477b O, 6672 0.6643 o.)61,,2 0.657 -1.1
0.522 0.663 0.66J3a 0.6572 0.6828 0.635o 0.659 -0.6o. 616j) 0.6665 0,6636 0!.485 0.656 -1. 1
o.66, 0. 609 u o 0 57Oc,,,a ( ý, ,1 C).R -5 k. ,3.,0. 5903) o. 6130 0.6 o096 o. 5921 o. 6oi -1.3
0.800 0.567 0. 5,•9 0. 53 56 0.'5701 G. 568 ]i O.557 ].8o. 5 4 6 53 b . 5716 0. 5679 0. 5485 0.559 -1.4
mU = slope 38.15 35.57 -30.76 52.39b = intercept
at (Wi 0 O) -1.71 +D. 35 +0,.20 -2.1.5Jo= intercept
at (Wi = 1) 36.hL 311.95 8o.9, 50.26
a. The upper number of each pair is the value obtained from the experimentalpoints.
b. The lower number of each pair is the value obtaincd frem the leastsquares line.
u3I
I
TABLE 5
EXPERIMLENTAL TRANSMISSION PROBABILITIESFOR EIGIIT CYLINDRICAL ORIFICES (SET II)
Orifice, K IOo( K-W)
L/R W Run 2 Run 3 Run 4 Run 6 K W
0.4462 0.819 0.0F7, 0.854 0.809 o.8i4 0.838 +2.-o.848b 0.811 0.7'JF 0.802 0.815 -0.5
o.6614 0.754 0.755a 0.752 0.754 0.751 0.748 -0.80.761b 0.742 0-724 0.731 0.740 -1.9
i.016 o0669 o.647a o.649 0.591 0.599 o.622 -7.00.680b 0.652 0.629 o.638 0.650 -2.8
1.291 0.616 0.614a 0.594 0.541 0.577 0.582 -5.5O.603b 0.598 0.570 0.595 0.592 -3.9
1.627 0.565 0.590a 0.555 0.497 0.520 0.556 -4.80.575b 0.541 0.510 0.523 0.537 -4.6
1.946 0.521 0.556a 0.484 0.4C2 0.486 o.497 -4.60.535b 0.49T 0.464 0.477 0.493 -5.4
2.263 0.465 0.508a 0.456 0.419 0.404 0.447 -7.6O.501b 0.460 0.425 0.437 0.455 -6.2
2.575 0.455 0.464a 0.448 0.414 0.446 o.443 -2.6o.469b 0.428 0.390 0.405 0.423 -7.0
m w slope 46.67 66.15 84.28 75.14b - intercept
at (Wj - 0) +1.24 -3.10 -9.07 -6.12Jo- intercept
at (Wi - 1) 47.91 63.05 75.21 67.02
a. The upper number of each petir is the value obtained from the experimentalpoints.
b. The lower number of each pair is the value obtained from the leastasqarea line.
64
TABLE U
EXITERIMENTAL TRANSY I S SION IPIOBAilITILSFOR EIGHT CONICAL ORilFICES (SElT III)
Orifice, K - ___(_-W)
L/R T, dog - W
.-05O 222.0 O.88) 0.87'(a C, -` 5 0 0.8)140. 68( b 0. 86 5 0. 886 -
3.889 26.8 0.884d 0.944 2a O.860 0.901 +1.80.886b 0.88 8 0.885 <0.1
1.005 18.y 0.8665 0,832a 0.956 0... +2.00. 8u6b 0.862 0.864 -0.5
0.421(, 0 0.8252 0. 870a u. 8 57 0.8614 +),.f0 82yb 0.825 0.82P +0.1
J4.051 15-2 0.7375 0.620a 0.5,81 O.oO0 -28.70.742b 0.756 0.759 +0.2
2.1835 9.65 0.6893 0.87ha 0.829 0.853 +25.6o.695b o.686 o.692 +0.3
3.857 9.15 0.6 187 C. 568a 0.595 0.582 -6.0o.625b 0.617 O.62i +0.3
2.072 0 0.5059 0.505a o.493 0.502 -0.80.515b O.5J 4 0. 510 C +0.8
m - slope 57.9-" 35.88b - intercept at (Wi-O) +0.68 -0.17JO= intercept at (Wi-1) 58.65 35.71
a. The upper nurnber o" cach pair is the value n Cpoints.
b. The lower number of each pair is the value obtained from the leastsquares line.
65
Ly'pi'-.a a Slecnt~ft between IlicasUted vapol pre'Ssu S~i reported by two laho,:rat cries,
IIOth Of Which USC Supp~losedly t he saiiie Knudsen effusion tcchnique, and is
b~Ctter: Lhnu the reprvuducibiIi ty in vapor pressure mtuasurcfments often Obtai ned
within a given laborat-ory. It wou~'d tlicrcfcrc appear thaLtLhe source of
thlese discrcpancics in vapor preSSUrL incasu,:cments may lie in phetnomena other
thtan flow of gas through an orifice.
Concurrently With our work, Macur, Edwards, slnd Wahlteck 4 .1 have attempted
to circu"-,-nt this probleiu Ot poor reproducibility in Knudsen effusion measure-
ment~s by using a Multicell technique in Whiichl thc em1phasis iS onl the aimul-
tancous determination Of several vapor pressures at the Sanme temp-erature; the
average deviuajo: in the pressures .calculated from the Several Cells in a
given run range from 1. to .3for indium~l and from I to 81 or gallium. The
agreemý-ntA from run to run (i.e., With chan~ge in temperature), as reflected
in the calculated Va1Lues Of AW,0,o of vaporization, is equally impressive;
the uncertainty quoted is ±0.10 Kcal/mole.
SECTION V
THLE MIlKER TECIINIQ11E
The Miker techinique for deicroil nation Of vIIpor pIeeSSures and Of molecular
weights of vapors at hig;h temperatures was described in detail !n Part Il" .
In this technique a vacuum microbalance is used tO obtain bothI the rate
of effusion from a suspended Knudsen cell and the recoil force which is
excrted oil the cell as a result of effusie..
The results reported in Part Il' dewonstiatecd the general validity
of the techn~que, and also demonbt rated the needI for vax ions re ii netuni s
if tile techniqiic was to provide a significant imp~rovemen~lt over tYp~ical
Torsion-Knudsen effusiun-ReA I data. Tile various refinemenlt-ls will bi:
rep or ted.
A. MODiFICATiONS TO APPARATUS
The early results 21 with the Miker appAratus indicated two major de-I
ficiencies: (1) When powcr to the furnace Was terminated in the courbc
oi meesuring the recoil force, the furnace cooled so slowly that rn u~n-q
desirably large correction for effusion during cooling was required.
(2) Manual control of the vacuum microbalance was satisfactory for rate
of effusion mcasurements, but not for recoil force measurements; aot~omatic
balance control appears to be a requirement for valid recoil force measure-
ments. Most of the modifications to be eescribed were made in thle course
of overcoming these two aeri4ciencies.
1. Furnace and Power Supply: Rapid cooling by thuý furnace appcared
to be Moat easily achieved through thie use of elements with very low
heat-capacity and with high surface area. Tunigsten muesh furnace
elements were considered butL rejected in favor of ýVaf)Iite LBape for
the following reasons: (1) Graphite tape retains its flexibility
C1lilpIeri It l ICtS Wk.''10 11t , ct C an I . a' ! it LiJ I SO 5hih tha vaL po r rooat ionl
1i 0i1 t ii' 1t))piii L,2 cI nmt'L apS d a I)rob1 1iii ; (3) Furnaice falbricil trol
insji cti Ae ',It I S r L''3d Iy aWCOU11p IiSlIe ill our shllpj) (4) The 1rt'siS-
. i vrty' of Crpti is so h i~ltha olne may easi Iv falriteate o furnace
X!i lj opemrateIs ~Ii frt( ,1y traom ai SCR- :nm en]I led , I bOAA, I I0k', bIil~ lýIine;
pore1 ;It at Lilavlae-:ar level is noel' more1 e~ý.i ii N andlt:!d anid
inlt roduLced ni aU theI) VflCIl111lll Le5tIII t hat[I i ,, it' IILIOwe r ti Dag, - veryI)
high cutrent levelts required by metaOl io heat'In', elements.
111e fu~rnace h~as a eyliIIdr.ica I ea~nfigerat ioui and eoonsiatab of six 13-cm
eiigthris of I .27-en- wide. p r'apluto tape lie Id at. t~i opaiaj butt n by maeLIi rued
graphite virings vsith pro pi13t iti set ews . ileLIocr r ica y , the three lengtihs
of t,1ape wh i C formf on1l10 le I ud'IA are7 illar I as aeLii'. othel
th t tit' two set,,, ot t.hreet ;1ie ti. h Ti tiV t't'i.'S (16'~e ('d IC rl i 2-
raly h the bot-tom gpitering. 3'hie res5LrtaCC' Of tilt' furnAIce is
-I X6 Ami aIt W C, t th no ia rat( i~oi sli e I ding, za current,1 oi -5OA rilsA
1) rod uces a tecmpera tture of 13000 C in1 . CC1 a c ll Supen~d*e inl t 1e C011Le
aof the: MI urn cs. V/eIower to Lt lie. I urnsý ie j ii L ertii fI P t-e-d , tie s kisilOndo 0
cell clool S inl lal at aaI ý rdt a of - 2 5-30 dIeg C/sec.
As, inticl tate a hove,, power toa the furnaýce li contr 1 ,' led withi a
single. I10A iliulcltroldeca ertireW gýate of Which is driven
by a ''Silicontrol'', oulse: gene ratilq uIi It
Tilt' other urnits tl the( contrlY0 100p are a tIlogStel'irienluli tietr-
moe oupl,, wh bi.cih resaponds to a ia oge s ir, furnace teicipera :. re, a stab In'
varxable re.ll'rene,' vantage, an elect roilic nlull detc~tac whlich resporids
to ainy error voA!Ltage between_ tlhe tblerliacaupl1e and the! reference voltage,
and a Coat cal 1e wh:ci' accepts t-he ariLpirt of the null. dc ect:rrr, provides
proport ionl)l, iie~rivat ive , 0110 integral cant rot actions, aInd riuppfles orn
opi apr ja e dn current to d~rive tlhe SiIi cantra(l uniit . Thle thermocouple
is used only for cont rol ; t~L~iU~ rtoircs arecr:sie with ail 6ut oca tic prhotoa-
elect~ric 'HIiOLOVulit ic'' yoree'
2. Automatic Control of Mi croha lance: Several changes hove been madc
in thle aut omiatic control system, aIl though Liu: basic cont rol loop
(Figure 20 in reference 2 1) relam inIS hef. same . The magneItl i C ampli L f icr
Which Suppli ed current to tlhe compensat ing coilIs has bee~n replaced
with a we 11- egolat~ed dc power Sul-ply and all emlitter- follower circuit
whi-in is driven by thle output of the CoIntrOller. This substitution
eliminated a significant ripple voltage from thL coils.
The "Photopot" light beam posit ion bensor 21has been replaced
with one of liter manufacture, which is wore stable and Which Canl
he readily operated inl high vacuum. The ne(w 'PhOtOFoL" Wasi. mounted
inside the balance chamber about 6.3 CM frOM theIL mirror affixed to thle
center of tile ha lance beam. The light beam originates , outside the
vacuIum system, in a 50W projector lamp, is defined by a 0.010-cm sliL,
reflected through a lens and window into the balance. chamber and ont-o
the balance mirror, and focused onto thle "Phiotopot_'. The output of
the projector lamp is monitored and controlled by a modification oi
the circuit described by RosenthaI4 Power for the 'Phot~opot" bridge
Circuit is now supplied by mercury batteries.
The weight change whiich can bv compensated by change in Current
throu~gh the ha lance coils (from 0 te clO00M) is 28mg. To avoid1 ~ opening the vacuum systeml to re-tare the ha lance after each 25-30mg
weight loss, a taring mechanism operable from outside the vacuum
system Was installed - A bel lows- sea led slinft d rivcn ocutS ide thei
vacuum chamber by a micrometer screw, carries four _,20-mg ring Weights
formed from Alumel wire 0.020 cm in diameter. As thle mictometer screw
is rotated, thle shaft is translated downward (upward) and the four
ring weights are sequentially pIaLcC on (removed from) a cross-arm
at tach~ed to the balance beam.
3. Miscellaneous Modifications: In the Miker technique effusing
vapor is directed downward, directly toward th-e window/prism thlrough
whlichi opt icalI pyromecter measurement s arc modeý. A shutte~r prot cets
the window during 'Lbe Interval bet~wcen pyrometerl reading; evL-n so,
the window acquires a significant de,.posit rathler rapidly. To a vL)id
e,e ning the entire vacuum syst em toe the otmosphecre whil c the wfndowI/
69
prism is rcmotvcd, cle,.ed, and Icplaccd, a very thin; bellows-
sealcd gate, v•oivy uas installed in thtic window mount and between the
window/prisr, ittself -nd tl',e furnace cl.'sbcr. 'The valve is a mod fi-
cat ion of the des igi reported b N ftSI te , and has perlormed
sat isfactori ly.
The effect of building erid equipment vibrattions on the micro-
bholance has been minimized by mounting the vacuum system on a steel-
reinforced concrete block (2x2x3 ft. high) which was cast in place
on the basement flooa; mechanical vacuum pumps in the vicinity have
been mounted on vibration-absorbing pods, etc.
For increased pumping capacity, there has been installed in the
vacuum system a larger diffusion pump (PMC-720) and a liquid nitrogen
trap (Cryo-sorb).
4. L, usioius Recoil Force s in measurements of rate of effusion and
of recoil force made immediate]y after tile modificaLions described
above were effected, the measured recoil force was much larger (e.g.,
by a factor of 1.5-2.0) than exp:ected from the rate of effusion
measuremcr,. s A ma jor source of the spurious portion of the recoil
force has been identified as a, interaction between the fields
su.:urrounding the furnuce power !eads and the magnet attached to the
belamer beam. This interaction has been minimized, and all but elim-
inated, by arranging the furnace power leads and the furnace elements
themuelves in a configuration symmetrical with respect to generation
of spurious iagneti c fields In, the balance chamber.
A second ma jar source of spurious recoil force was inadequate
shielding of the "Photopot" from stray light in general, and specifi-
cally from light from the furnace. Str-ay light from the furnace
caused a shift in the null point of the "Photopot"; when the furnace
power was terminated to measure the recoil force, the stray light
from the furnace was also terminated. The nail point of the "Photopot"
shifted bak to tLie "furnace cold" position, but since this shift
coincided with a recoil force measurement, the shift was incorporated
into the apparent recoil force. Shielding of the "Phot.opot" was
relatively simple, once the difficulty had been identified.
70
The residual spurious forces which remain appear to arise from
pressure-dependent phenomena, e.g., thermomolecular flow forces4.
However, these resldual spurious forces now constitute only -1- of
a typical expected recoil force, compared with --50 to 100% before
the various corrective actions were taken.
B. EXPERIMENTAL RESULTS
We have only recently eliminated (or minimized) the spurious recoil
forces discussed above. The experimental data we report were obtained
with the modified furnace and automatic control system, but before the
spurious forces were identified and eliminated, and are presented to
indicate the precision which has been obtained in recoil force measurements
with the microbalance.
The data in Table 7 for the vaporization of tin were obtained with a
graphite Miker cell2l the conical orifice of which is described by T -
28.80, (L/ro) 4.98, W = 0.908, and f = 1.088.
The results of Table 7 are typical illustrations of the reproducibility
obtainable with the Miker system. It should be noted in particular that
the reproducibility in measurements of recoil force is at least as good
as the reproducibility ip. measurements of rate of effusion.
We expect to observe similar reproducibility and good agreement
between effusion and recoil measurements in subsequent vapor pressure
deter-ainations.
71
TABLE 7
HMiRM DATA FOR VAPORIZATIONOF TIN
Rate of Recoil Recoil Preusure1 -. din/cm2
Temp, effusion, Current, Maoaui
1466 1.061 0.306 l00.4 11.2 21.7
1470 1.078 0.310 101.7 11.4 22.0
1473 1.138 0.298 97.7 12.• 21.1
1473 i.o67 0.302 99.1 11.4 2i.4
1486 1.156 0.328 107.6 12.3 23.2
1489 1.070 0.327 107.3 11.4 23.2
SRecoil mass is that mama which, under the acceleration of gravity,counterbalances the recoil force; balance calibration = 0.328 mg/mA.
t "at of eff.,1e.AF.. ,gi~..o., P from rccoal
The "accepted" value45 is 7.8 dcy/cm2 at 14800K.
72
U
I
SECTION VI
CALORIMETRIC STUDIES OF
VAPORIZATION PROCESSES
Determination of equilibrium vapor pressures and/or decomposition
pressures, with subsequent calculation of enthalpy of vaporization and/or
decomposition and of enthalpy of formation of the gaseous species evolved,
occupies a key role - one is tempted to say the keystone role - in high
temperature chemistry (IITC). From the various measurements of (supposedly)
equilibrium pressures in various laboratories have arisen numerous dis-
crepancies: between Knudsen eifusion data and mass spectrometric data;
between Knudsen effusion data and Langmuir vaporization data; and even
between two sets of Knudsen effusion data, both of which were obtained by
reputable workers using, supposedly, the same technique. Attempts to
resolve these discrepancies have resulted in, among other things, the
formation at the 196.6 Gordon Research Conference on HTC, of a committee
to study establishment of vapor pressure standards above 500°K and below
10-3 atmosphere.
In addition to refinement and standardization of effusion techniques,
an endeavor to which the previous sections of this report is devoted,
new approaches are needed. A more or less obvious one, except for
experimental difficulties encountered at high temperatures, is direct
calorimetric measurement of enthalpy of vaporization and/or decomposition.
Sunner and MorawetZ46 (at the University of Lund, Sweden) 1.,;e studied
intensively the problem of calorimetric measurement of heats of vaporization
of various hydrocarbons and other organic compounds at 25'C. They have
successfully measured heats of vaporization with a precision and accuracy
better than 0.1 kcal/mole for materials with vapor pressures as low as 10-4
torr at 250C. They have also measured differences in heat of vaporization
as a function of the effusion orifice geometry.
I.)
N
In conjunction with work supported by another contract we became
interested two to three years ago in the possibility of direct calorimetric
measurement of enthalpies of decompositi'Ln, e.g. A(,) - B(s) + C(g). Our
initial approach to the direct measurement of enthalpy changes was essentially
to copy the furnace and calorimeter configuration of Speros and Woodhouse's
'ýquantitative differential thermal analysis" (QDTA) system47 and to make
various changes in the electronic control circuity and the output sigial.
In evaluating our QDTA system , we measured enthalpies of fusion of several
metals with a precision of 0.5-1.5% and an accuracy of 1-5% in the range
150-4500 C. Subsequently we investigated the decompostiion of PbCO3 , ZnCO3,
and NH.4C1; with QDTA we were able to determine with reasonable precision
and accuracy the enthalpy change for these decompositions, including the
change for each of three steps in the decomposition of PbCO:
Our Pxperience with the QDTA concept has led us to make several sig-nificant modifications in the furnace-calorimeter configuration and in
the control-output circuitry; we refer to the new system as "Differential
Scanning Calorimetry" (DSC)4 9 . These modifications are'presently (Summer,
1967) being debugged and tested. We expect that with this new DSC w-
shall be able to measure in the range 30-1000%C, endothermic enthalpy
changes resulting frcA fusion, vaporization, decomposition and phase
transition, and perhaps even exothermic enthalpies of reaction.
It is our further opinion that it is feasible, within the present 4
state-of-the-art, to design and build a DSC system which would provide
direct measurement of enthalpy changes in the range 1000-2000*K, and
perhaps to 2500'K. Such a system would make available to "higher" tem-
perature chemists the capabilities presently available in commercial
equipment4 9 up to 50 0 C and expected in our DSC system up to 1000C.
Another virtue of the DSC technique is thist with the one assump-
tion that the rate of energy input to the calorimeter is proportional to
the rate of the (endothermic) process occurring in the calorimeter, the
data one obtains are directly interpretable as the rate of the process
as a function of temperature and of time. It is then rather straight
forward47 P5 0 to obtain rate constants over a range of temperatures, and
hence the activation energy, from a single 1-to-3-hour run. With the
older QDTA system we have obtained•a energies of activation for the various
74
steps in the decomposition of rbC03 and for the homogeneous decomposition(in solution) of a complex organic azo compound.
Tn Suinary, we aie CUnvinced that direct caloriTretric measurement of
enthalpies of decomposition, vaporization, etc., are presently achievable
to 1000'C, and with a relatively modest development effort could be achieved
at 20000 C. While these techniques are not likely to replace effusion
techniques in the near future, they can provide for high temperature
processes data obtained by other-than-equilibrium techniques; such supple-
mentary and complementary data are sorely needed in many high temperuture
chemical systems.
7!
TABLE 8
IDENTIFICATION OF MAJOR COMMERCIAL COMPONEETS OF APPARATUS
Item Suppli.e•
Amplifier, Leek-in Princeton Applied Research Corp.PAR JB-5 Princeton Junction, N.J.
Controller, C.A.T. Leeds and Northrup CompanyType 10877 4901 Stenton Avenue
Philadelphia 44, Penna.
Lamp, Projector Local Photographic8V, 50W, Type 13113C-04 Supply ShopPhilips (Holland)
Null detector, d.c. Leeds and Northrup CompanyType 9834-2 4901 Stenron Avenue
Philadelphia 44, Penna.
Photopot Giannini Controls Corporation55 N. Vernon AvenuePasadena, California
Power Supply, Sorensen Products30V d.c., QB28-i Raytheon Co.
S. Norwalk, Conn.
Pressure Meter, Trans-Sonics, IncorporatedEquibar Type 120 P.O. Box 328
Lexington 73, Massachusetts
Pump, diffusion, oil, Consolidated Vacuum Corporation4 in., PMC-720 1775 Mt. Read Blvd.
Rochester 3, New York
Pyrometer, optical Pyrometer Instrument Co.Photoelectric, "Fhotomatic" Bergenfield, N.J.
Valve, vacuum, Cranville-Phillips Companyvariable leak 5675 E. Arapahoe AvenueCat. No. 9101-M Boulder, Colorado 80301
Silicontrol Pulse Unit VecTrol Engineering Div.Type VS6332AF Sprague Electric Co.
P.O. Box 1089Stanford, Conn.
Trap, liquid nitrogen Granville-Phillips Co.yu-•t, l, 5673 E. Arapahoe Ave.
Boulder, Colorado 80301
Tuning Fork American Time ProductsType 40 61-20 Woodside Ave.
Woodside, N.Y. 11377
76
RIEIRENCES
1. Mi. Knudsen, Ann..P-s. -2-. 75, 9)) (i)@)) ; "Kinetic Theory of Cases,Methuen, London (,1954).
2. J. L. Margrave, In "Physico-Chumical Measurements at Hitgh TempCistures"
(3. M Bockris, J. L. White, and J. D. MacKenzie, eds.). ButterworthsScientific Publications, London (15,59).
3. Various chapters in "Characterization of High Temperature Vapors"(J. L. Margrave, ed.). John, Wiley and Soils, Inc., New Yoark (1967).
h. By definition, an ideal orifice has zero length; if its radius is thesame as the radius of the smaller end of) an actual (conical) orifice,the orifices are said to correspond.
5. P. Clsusing, Physica 0, 65 (1929); Ann. Physik 12, 961 (1932).
6. W. C. DeMarcus, Technical Report K-1502, parts 1-6, Oak Ridge GaseousDiffusion Plant, Oak Ridge, Tennessee (195T); and W. C. DeMarcus andE. H. Hopper, J. Chem. P:Ls. 2.ý, 1344 (1955).
7. E. W. Balson, J. Phys. CheQ,_ 6, 1151 (1961).
8. R. P. -zkows i, J. L. Margrave, and S. M. Robinson, J. Phys. Chem. a,5 229 (1963).
9. W. L. Winterbottom and J. P. Hirth, J. Chem. PhYs. 7, 784 (1962).
10. P. Clausing, Z. Physik 66, 471 (1930).
.11. R. D. Freeman and A. W. Searcy, j. Chem. Phys. 22, 762, 31' (1954).
12. R. D. Freeman, "Molecular Flow and the Effusion Process in the Measurementof Vapoz Pressures", Technical i.;port ASD-TDR-754, Part I, 1965(AD42314O).
13. R. D. Freeman and J. G. Edwards. in "Condensation and Evanoration ofSJulids" (E. Rutner, P. Goldtinger, and J. Birth, eds.); Proc. Intnl.Symposium, Dayton, Ohio, September, 1962. Gordon and Breach, New York(19614). p. 1-1.
1h. J. C. Edwards, Ph.D. Thesis, Oklahoma State University, Stillwater,
1964 (AD469505).
15. J. G. Edwards and R. D. Freeman, to be publi'hed.
16. A.-endix E of referencc 14.
17. D. H. Davis, L. L. Levenson, and N. Milleron in Rarefied Gas Dynamics,(L. Talbot, Ed.) Academic Press. New York (1961). p. 99.
18. E. M. Sparrow and V. K. Jonsson, AIAA Journal 1, 1081 (1963).
77
Ii
1 A. NASA Technical Notes, Lewis Research Center, ClevelAnd, Ohio
a. E. A. Richley and C. D. Bogart, NASA TN D-2115 (February, 1964).b. E. A Richley and T. W. Reynolds, NASA TN D-2330 (June, 1964).c. 11. Cook and E. A. Richley, NASA TN D-2480 (Septembey, 1964).d. T. W. Reynulds and E. A. Richley, NASA TN D-I564 (October, 1961t).e. T. W. Reynolds and E. A. Richley, NASA TN D-3225 (January, 1966).
20. J. W. Ward, "Use of the Knudsen Effusion Method, A Literature Survey"
Report LA-3006, Los Alamos Scientific Laboratory of the University of
California. (Available from the Office of Technical Services) May, 1964.
21. R. D. Freeman, "Molecular Flow and the Effusion Process in the Measurementof Vapor Pressures," Technical Report ASD-TDR-65-754, Part II (AD612953).
22. N. F. Ramsey, "Molcular Bemus", Oxford University Press (1956).
23. W. L. Fite and R. T. Brackknau, Ph2ysicl Review ]1?, !141 (1958).
24. T. H. Batzer and R. H. McFarland, Rev. Sci. Instr. I, 528 (1965);
T. H. Batzer, private communication.
25. J. C. Sheffield, Rev. Sci. Instr. Lb, 1269 (1965).
26. J. Q. Adams, Ph.D. Thesis, University of Illinois, Urbana (1961).
27. K. C. Wang, Ph.D. Thesis, Illin3is Institute of Technology, Chicago (1966).
28. F. C. Hurlbut, Report No. AS-66-10, College of Engineering, University
of California, Berkeley, August, 1966; paper presented at Rarefied GasDynamics Fifth International Symposium, Oxford, England, July, 1966.
29. V. Ruth and J. P. Hirth, (same as 13). p. 99.
30. J. W. Ward, "A Study of Some of the Parameters Affecting Knudsen Effusion",
Report LA-3509, Los Alamos Scienti fic Laboratory of the University ofCalifornia (Available from CrSTI). Also, Ph.D. Thesis, University of
New Mexico, Albuquerque (1966).
31. P. W. Gilles, Ann.Tl. Rev. Phys. Chem. 1., 355 (196].).
52. K. D. Carlson, Ph. D. Thesis, University of Karsas, Lawrence (1960);Argonne National Laboratory Report ANL-6156 (1960).
35. J. D. McKinley, Jr. and J. E. Vance, 7. ghem.. Py. 22, 1120 (1954).
34. K. Motzfeldt, 1. Phys. Chem. 52, 139 (1955).
35. C. I. Whitman, Ch. e.m. Phya. ?C, 161 (1952).
36. A. J. Boyer and T. R. Meadowcroft, Trans. Met. Soc. AI , 388 (1965).
(0
4
.1 37. K. D. Carlson, 1'. W. Gilles, and R. J. Thorn, J. Che.a. I' , 225
(1963).
38. C. Hall, Rev. Set. lnjtr. 5, 1)1 (1962).
39. A. L. Ball, Jr., MS Thesis, Oklahoma State University, Stillwater (196'3).
40. R. D. Freeman and R. E. Gebelt, unpublished resujlts.
41. C. J. Macur, R. K. Edwards and P. C. Wahlbeck, J. V_.. Che.. In,
2956 0.96-6).-
142. N. C. Mushovae, Technical Paper TP-63-11, Sprague Electric Co., NorthAdams, Mass. Figure 8.
143. L. A. Rosenthal, Rev. Sci. Instr. L01 1529 (1965).
0h1. First three papers in ",Vacuum Micxobalance Technique", vol. 5, (K. It.Bchrndt, ed.) Plenum Press, New York (0966).
45. R. Ilultgren, R. L. Orr, P. D. Anderson, and K. R. Kelley, "SelectedValues of TheunodynaMic Properties of Metals and Alloys". J. Wiley andSons, Inc., New York (l96.).
46. S. Sunner and E. Morawetz, Act,-Chem. Scand. 2-., 13 (i)63); privatecolmlunication, June, 19t,(,
147. D. M. Speros and R. L. Woodhouse, J. PLhys. Chetm. ýJ, 26164 (1963).
48. P. D. Gwinup, Ph.D. thesis, Oklahoma State University, Stillwatet (1967).
49. Pe-kin-Elrcr Cotp., Norwalk, Connecticut, Differential Scanning Cal-orimeter DSC-113.L. S. Watson, et al., Anal. Chem. •, 1233 (59Q4).M. J. 0 Neill, Aak. Chew. 56. L?3. (1964).
50. 1i. J. Borchardt and F. Daniels, J. Am. Chem. Soc., L.), 41 (1957).
7)
UNCL.AS;jIFL. E
3e aiy C as[c to DOCUMENT CONTROL DATA -R&D(S~cuui.1 cI... iftarl_. 1Ito -y r ,eefrt *s,4 tndeorng annotaitan must be *nIorod when ih. overall rop.lr in ci-t.di )
[IOrnIGINA 7IN G ACTI'-IT Cf-t Autor 1-0 RLPcOT EEUR TVQ C ;;Fi:
Oklahoma State University UrNCI.SSI FIEDI,ý,t.of Chemistry -1 GROUP
SIi II wa re r, 0!(la h oma 74074 ___ __ _________
_377-LCT IITL E
"ýi ')occil ar Fl ow and the Effusion Process ill the Mtasuremntuoi Vapor Pressuros'
_F ESc C771PT I VE oIN 0 (7
-po a I rej i at, d in, Jn v O. I*..
11101. Re~port, I Septcmber 1964 to 31 Moy 1967
Frceman, Robert D).
6 METPO FT 0;ATi 7a TOTAL NQ OF PAqFP =79- NO Or RLF#
NovIHILO ., 1967 ________ 79 50 ______
8. CONT11ACT OH -RAN~rNO~ SO ORIGINATOR's REPORT1 NUMOICft(S)
AF --3(657)-8767b. 'AOJCC- NQ 7360 ASD-TDR-63-754, P't:. III
Task: 736004 ________________________
Ob. OTHIC.9 K PORT tdO(S) (Any thonu- ove 9--may be samIned
IS ~ ILASLIT/LMITTIN N11~5 This documlent has been approved for public release
andl sales; itS distribution is unlimited.
I I SLIPPL.EMIEN IAMY NOTES la SPON5OIIING MILITARY ACTIVITY
AF Materials Laboratory (MAYT)Wright-Patt~erson AFB, Chio
3i AUSINAC TOr extensions to conical orifices of Claiising's analysis of angular dis-
LribUt-oln of' mol~eculetu effusing from cylindrical orifices has resulted in numericsvalues for transmission probabilities and recoil- ffirre correc.tion factors which aretLbulated. With these results, it is demonstrated tha-t the optimum orifice geonm-etry for (1) recail force measurements is a diverging conical orifice with semi-apex angle of 30c, (2) delivery of ma,:cimum fraction of cffusing molecules onto (onlinto) a targct (,aperture) is a long cylindrical orifice.
Modifications to the angtliar distribution apparatus are described and experi-
mental results given for four orifices and two gases over the pressure range 5 to900 dyn/cm'2. The mostý incerestitag aspect of the results, one apparently not previ-
oaisly noticed in angular distribution results, is the presence of maxima and minimi
in plots of Li VS. £, where 6 is the (experimental value - theoretical value) oiP?,, the fract 'ion of effusing r-olecules which flow per steradian at angle @ from
Leorifice axis, These rnaxlr.:& and minima have been correlated with the relativecontribution from the orifice well to the total fIlux at angle P.
Additional results for expctillental transmission prol abilities of orifices de-
teruirined by the Multicell technique are generally in agreement with theoretical
values within 2 tQ 5%.The Miker technique for simultaneous determination of vapor pressure by rate -)f
effusion and by recoil force measurements has been refined to the point that recoýA
force data are as reproducible as rate of cffusion measurements. Several sourcesof' Spurious recoil force have been identified and eliminated, A new furnace and amodified- automatic control system for the microbaiance are described.
D D I A44 1473 ____UNCLA SSI FIEDSecurity Classification
UlNC LA SSI F1' I'DScict I Iy (IasItfI catliOn
4 Lt, A Ls 1 Li LINK1 CKLY VORDS - _
Vapor: PrcssuruMo]ccular Flow1 ranstttisj ;on I'roboibilifly for Orif ices;Mit CIo hl a 1a1ce
* Molecular Deatiis
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