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INDUSTRIAL AND ENGINEERING CHEMISTRY
edge meanwhile moving inwai d. The effects of such changes on the dimensions of the compact bring to mind the familar picture, drawn by Drapeau (16) and others, of axially prepressed hollow cylinders shrinking radially while simultaneously increasing in length. Lack of data concerning the pore size distribution in Drapeau’s experiments prevents any quantitative treatment.
Vol. 40, No. 5
for such densification is primarily temperature-dependent and the time-temperature relation can be deduced from the coefficient of viscosity. The viscosity, in turn, is related t o the phenomenon of self-diffusion, in accordance with preliminary experimental results.
Literature Cited
Conclusions One aspect of the problem of sintering is dealt with in this paper-namely, the mechanism of the changes in volume taking place in the powder compacts. The theory, suggested by Frenkel’s work (Q), is developed here for application to several actual cases and accounts for experimental results found here and elsen-here. Previous attempts to explain the phenomena of sintering have been obscured by an improper assessment of the role of such transient effects as recrystallization and the desorption and expulsion of gases from the metal during heat treatment of the powder compact. iiccording to the theory presented here, sintering is attributed to a viscous flow of metal under the influence of surface tension, modified by a gas pressure. According t o the calculations, in compacts containing a range of pore sizes, the finer pores shrink before the larger ones. If a gas exists under pressure in the pores, the first activity is the shrinkage of the finest pores to a stable size independent of temperature or further heating time; later, larger pores become active, but those larger than a critical size expand instead of shrinking, and also reach a stable size. I n the ideal cases where no foreign gas is present or where the gas can diffuse out through the lattice of the metal, complete densification of a powder compact would eventually occur below the melting point of the metal. The time required
(1) Balshin, M. Y., Vestnilc Metalloprom., 16, 87 (1936). (2) Chalmers, B., Proc. Roy. SOC.( L o n d o n ) , 156, 427 (1936). (3) Delisle, L., Trans. Am. Electrochem. Soc., 85, 171 (1944). (4) Frenkel, J., J . Phys., U.S.S.R., 9 , No. 5, 385 (1945).
(5) Gibbs, J. W., “Thermodynamics,” Vol. I, New York, Longmans,
Green and Co., 1931. (6) Hfittig, G. F., Kolloid-Z., 97, 281 (1941). (7) Jones, W. D., “Principles of’PowderMetallurgy,” London, Edward Arnold & Co.; 1937. (8) Kanter, J. J., Metals Technol., 4, 8 (1937). (9) Kingston, W. E., in Seelig, R. P., Seminar on Pressing of Metal Powders, Metals Technol., Tech. Pub. 2236 (August 1947). (10) Philipps, H. B., “Vector Analysis,” New York, John Wiley & Sons, 1933. (11) Khines, F. N., Trans. Am. Inst. Mining Met. Engrs., Inst. Metals Diu., 13, No. 5, 474 (1946). (12) Samoilowich.A,, Acta Phgsicochim., U.R.S.S., 20, 97 (1945). (13) Seits, F., “Physics of Metals,” New- York, McGraw-Hill Book Co., 1943. (14) Shaler, A. J., Sc.D. thesis, Dept. of Metallurgy, Mass. Inst. Tech., June 1946. (15) M‘ulff, John (Ed.), “Powder Metallurgy,” Drapeau, J. E., Chap. 31, Cleveland, American Society for Metals, 1942. (16) Ibid., Libsoh, Volterra, and Wulff, Chap. 35. (17) Ibid., Wretblad and Wulff, Chap. 4. (18) Wyman, L. L., and Kelley, F. C., Trans. Am. Inst. Mining M e t . EnQrs., I n s t . Metals.Dh., 93, 208 (1931). RECEIVED November 28, 1947.
Trajectories of Heavy Molecules in Air R. B. Jacobs1 and S. F. Kapff DISTILLATION PRODUCTS, INC., ROCHESTER, N . Y.
’
Measurements of the trajectories of several commercial pump oils in air at about m m . of mercury indicate that the oil molecules are only slightly deflected when they cdlide w-ith air molecules. This fact is in agreement with experience in the use of oil diffusion pumps where the oil jet traverses a distance of many times the mean free path of the oil molecules without being appreciably deflected. Theoretical considerations indicate that the oil molecules su@r small deflections because of their relatively large masses and because they do not act as cohesive bodies during impact. I t is probable that the air molecules strike only one or two of the outermost hydrogen atoms on the oil molecule. The hydrogen atoms rebound freely on impact and later transfer this newly acquired momentum to the oil molecule as a whole. This mechanism provides the oil molecule with a certain springiness which results in a very small transfer of momentum on collision with an air molecule.
IiY
THE design of certain types of high vacuum equipment, it is often important to know how far a heavy oil molecule will travel a t a given air pressure without appreciable deflection. 1
Present address, Standard Oil Company (Indiana), Chicago, Ill.
For instance, in an oil diffusion pump, a requirement for successful operation ip that an appreciable fraction of the oil molecules traverse the distance between the nozzle outlet and the condenser wall without significant deflections from a straight-line path. When the oil jet is unable to reach the jvall because of excessive air pressure, the pumping action of the jet ceases. Also in molecular distillation, the object is to condense the evaporating molecules as rapidly as they leave the evaporator. This is achieved in practice by exhausting the air sufficiently to ensure almost rectilinear paths for the oil molecules between evaporator and condenser. At higher air pressures, oil molecules may be deflected sufficiently by multiple collisions with air molecules to cause their return t o the evaporator before they reach the condenser (to the obvious detriment of the process). In both illustrations, successful operation is dependent on the ability of the oil molecules to travel a definite distance without being deflected from their initial paths more than a certain amount by the air molecules which they encounter. Xow, from kinetic theory if the size of the oil molecules is known (from electron diffraction data), the average distance which an oil molecule travels between collisions with air molecules can be computed at any air pressure. This distance is known as the mean free path of the oil molecules in air a t the given‘pressure. But because the oil molecules are much heavier than the air molecules, and because
May 1948
INDUSTRIAL AND ENGINEERING CHEMISTRY
they possess a certain springiness, they are not greatly deflected i n a single encounter with a n air molecule. Accordingly, for practical purposes, the distance which a n oil molecule travels without serious deflection is many times greater than its mean free path. The mean free path is commonly used as a rough criterion for rectilinear travel, although practice in the operation of high TARGET vacuum equipment has 2ND SLIT shown it t o be a far too conservative design basis. IST SLIT The experiments described THERMOCOUPLE here were directed toward BOILER a quantitative basis for arriving at the distance which an oil molecule Figure 1. Cross Section of Molecular Dew Apparatus can in air at a Showing Essential Elements given Pressure without exceeding a given deflection. For the design engineer working with oil diffusion pumps or on molecular distillation equipment, the results as given in the first two sections of this paper probably will be sufficient within themselves. However, from a theoretical standpoint i t was felt that the present results had a unique interest inasmuch as the work was with molecules of known dimensions from which mean free paths could be calculated directly. From measurements on the deflections of the oil molecules the momentum changes associated with each collision could be calculated, and finally, information could be obtained regarding the approximate nature of a n air-oil collision. Previous workers in this field (2, S) generally employed metallic atoms or other 'lighter molecules in their scattering experiments; hence, they relied on the indirect and far less certain method of using gaseous viscosity data t o obtain molecular or atomic dimensions. The fact that the authors were able t o use oil molecules of known dimensions gave a considerable advantage in theoretical interpretation. Scattering experiments were made on several commercial vacuum pump oils. Results indicate that the oil molecules were scattered considerably less than would be the case if they behaved a s hard spheres. This might have been anticipated from their known'structures, andit means that the effective free path of heavy oil molecules is considerably greater than the actual distances
.
843
traveled between collisions. This result, likewise, might have been anticipated by anyone familiar with the behavior of jets of oil vapor in vacuum systems. Analysis indicated that, in general, collisions between air and oil molecules must be partially inelastic because this type of collision was the only reasonable mechanism which accounted for the small observed momentum exchanges associated with each collision. The deflection which an oil molecule suffers on collision with an air molecule depends on the position of the point of contact with respect to the two centers of mass and on the physical nature of the collision process. The first factor is statistical and subject to mathematical treatment. The second factor depends on the physical properties of the two molecules involved-that is, if the oil molecule acts like a rigid sphere, the air molecule will transfer to it the maximum momentum consistent with momentum conservation. If, how-, ever, the oil molecule behaves like a loosely bound structure, the collision might be essentially one between a n air molecule and a hydrogen atom; in this case the momentum transfer will be considerably smaller. Now, if the number of collisions per unit time which molecule experiences is known (calculated from its collision area), and also the percentage of these collisions which deflect the oil molecule more than a certain amount is known (from experimental data), the average momentum tPansfer per collision can be computed. The average collision results in a momentum transfer which is small compared to that possible for hard spheres. This is consistent with the many possible rotational degrees of freedom of an oil molecule, and with its rather loose structure.
Experimental The apparatus (Figures 1 and 2) was the same as that used by the authors for the determination of vapor pressures of various oils. It was previously described in detail ( 1 ) . Essentially it consisted of a boiler, a system of collimating slits, and a polished target. At equilibrium the number of molecules arriving at the target is a function of the geometry of the system, of the boiler temperature, and of the number of effective collisions between boiler and target. The number of molecules leaving the target is solely a function of the target temperature. The procedure was first to determine the target and boiler temperature at high vacuum, case 1, then t o redetermine these same two qualities after allowing the air pressure to rise to approximately 10-8 mm., case 2. As the target temperature is the same for case 1 and case 2, the number of molecules leaving, and hence arriving, at the target is the same in both cases. I n case 1, the number of molecules starting out in the cone defined by the slit system is (semiapex angle = 1'40') : Ni
-
kPIT1-I"
(1)
where k is a geometric factor and PI and TIare the vapor pressure and temperature of the oil in the boiler. Now, whereas, in case 1 all of the molecules reach the target, in case 2, N 2 molecules similarly start out in that direction but because of scattering only Nl arrive there. N2 is given by a n expression similar to Equation 1:
Ns
-
kp~Tn-'/~
(2)
where the subscripts refer t o case 2. It then follows t h a t the probability, s, of a single molecule reaching the target is $
Figure 2.
Molecular Dew Apparatus Assembly
9
NI N2
(3)
It is t o be understood, of course, that s relates t o our particular apparatus operated at a pressure of 1 micron of mercury. Combining Equations 1,2,and 3,
INDUSTRIAL AND ENGINEERING CHEMISTRY
a44 Table I.
Oil Amoil Octoil Amoil-S
Experimental Determination of Fraction of Nonintercepted Molecules, s
No. of Independent Determinations 5 6 4
Pressure Range of Measurements, Microns 0.8-147 0.47-0.95 0.52-0.96
Corrected t o 1.0 hlicron Pressure 0.226 0.131 0.122
5,
Vol. 40, No. 5
Numerical values for u1 were obtained by constructing scale models of the oil molecules. For this purpose Fisher-Hirschfelder models were employed; the dimensions of these are based upon electron attraction data. The models were projected on three mutually perpendicular planes, and three areas obtained, from which an averaged u1is computed. Equation 6 is reduced to a more convenient form by substituting the relation
Table 11. Computed Free Paths in Air at 1 Micron Oil Amoil Amoil-S Octoil
Table 111. Oil Amoil Amoil-S Octoil
Molecular Weight 306.2 343.3 390.3
Free Path, Cm. (Computed) 0.601 0.474 0.482
where m and T refer to molecular masses and temperatures and the subscript 1 refers t o oil and 2 refers to air. Then,
Percentage of Effective Collisionsa, E s, Observed 0.226 0.131 0.122
E 0.107 0.119 0.117
a Effective collisions result in deflecting the oil molecule o u t of a cone of aemiapex angle l o 40'.
(4)
from which s can be computed in terms of experimentally measured quantities.
Measurements. The experimental measurements involve the following: ( a ) the geometry of the system; ( b ) the temperature of the boiler and oi the target; and (c) the air pressure in case 2. The respective contributions to the experimental error from the above sources are estimated as follows: ( a ) rl%, based on accuracy of measuring slit openings; (71) =~10%, based on ability to reproduce temperature readings a t equilibrium to within 1O K., and (c) -lo?& based on estimated absolute accuracy of a n ionization gage (VG-la). Systematic error is contributed by (a) and (c), whereas random errors result from ( b ) . Usually a series of five or six determinations resulted in a probable error of -5% agreeing with the estimate given in ( b ) of rlOyo for a single determination. The error of -10% in the pressure measurement i s about what can be expected viithout extreme precautions of thorough outgassing, multiple gages, etc. ; these were not considered practical for the present experiment. Accordingly, the probable limits of error for the results given in Table I are =F15%.
Percentage of Effective Collisions Now computation can be made of the percentage of effective collisions, E-that is, collisions of sufficient violence to deflect the oil molecule out of the measured beam in our apparatus-by first compubing the mean free paths of the oil molecules, and then observing that if a fraction E of the collisions is effective, the beam will be attenuated according to the following equation: s = exp
-E1 X
Substituting numerical values in Equation 7, the mean free paths Xl,z are then computed (Table 11). The results of solving for E for the three oils using the values of X in Table I1 and employing Equation 5 are shown in Table 111.
Theoretical Determination of Collision Efficiency, E For a particular experimental arrangement, E may be computed if it is known what kind of collision is involved-that is, whether it be an elastic collision between two hard spheres or an inelastic one between two deformable bodies. Theoretical E's are computed in terms of the momentum transferred to the oil molecule by the air molecule; later, these are fitted into a reasonable collision process. Theoretical curves giving point collision efficiencies against path length of the oil molecule are computed for different assumed momentum transfers. The point collision efficiency (Figure 3) is defined as the percentage of collisions a t a given point along the path of the oil molecule which result in effective deflections of the oil molecule. The area under one of these curves divided by the total path length then gives the percentage of effective collisions along the path. The calculation of collision efficiencies for the purpose of drafting Figure 3 is somevhat laborious. This type of calculation would not be necessary for an experimental arrangement with a sharply defined molecular beam. The method employed is as follows: Spherical symmetry for the oil molecule is assumed for simplification. Spherical coordinates with origin a t the center of the oil molecule are used. The nomenclature is given in Figure 4.
(5)
where I is equal to the path length and h is equal to the mean free path of the oil molecule in air at 1 micron. For computation of X, the known dimensions of the oil molecules are used. of, The following expression ( 4 ) is used for the mean free path, h , ~ a molecule of diameter u1, moving through a second gas of molecular density n and average molecular diameter UZ:
-
DISTANCE FROM SOURCE ICMS.)
TARQET
I 2
where CI and C, refer to the velocities of the oil and air molecules, respectively.
4
p;1
10
Figure 3. Theoretical Derived Curves Showing Percentage Effective Collisions vs. Position of Oil Molecule along Its Path
INDUSTRIAL AND ENGINEERING CHEMISTRY
May 1948
f
DIRECTION OF 0 IL MOLECULE NORMA4 TO SURFACE
I
/
Aesumethat a direct hit a t (p = 0, e = 90") results in the average kransferaf linear momentum to the oil molecule of amount p , then a n oblique hit will transfer transverse momentum in the amount g cos .p&n e. The problem then is one of finding the percentage *of collisions which transfer transverse linear momentum t h a n s o m e arbitrary value, eb, where E is a function of patffg?? tion. For cdllisions occurring at a given value of 8, there is a n associated maximum a(am), such that only collisions for CY less than am areteffective. The ratio of the number of effective to the t o t a l number of collisions for a given e then is from the cosine law ?ofkinetic theory:
so Llm
Pi/*
= sin2 am (for 8 = constant)
a
J 0 con a x sin ada 'Now, ant is dehermined by the condition that the transverse imomentum transfer p cos a sin e 3 E p or that cos CY sin e 3 E , .where E is determined by the system geometry. Extending this to all collisions (for all e's) the fraction of colli:sions resulting in transfer of transverse momentum greater than ithe minimum required for an effective collision is $sin2
am ds E
*
.
$sine am sin 8dedgj $sin Bd8dg
where the indicated integration is over the surface of the oil molecule. Because of axial symmetry this reduces t o
"r sin
CY,,,
Now, experimentally, the effectiveness of a collision depends both on the transverse momentum transfer and upon the distance from the source of the molecule at the time of collision. (The effectiveness decreases linearly as the distance from. the source increases.) For the apparatus used, a n effective collision at the source must impart transverse momentum to the oil molecule at least equal t o 0.0289 times its linear momentum corresponding to the allowable divergence of the beam. Collision efficiency against path location was plotted for different p's (Figure 3). Now, by selecting a curve from the family indicated in Figure 3 to make E (calculated) equal to E (observed), the avera e transfer of momentums, p , for the different oils may be founpd. The results are given in Table IV.
Nature of the Collision Process
DIRECTION OF AIR
'Figure 1. Coordinate System Employed i n Theoretical Discussion
X sin ede
t h e lower limit of the integration is determined by the condition, mos,anr sin e 3 E .
845
The values of p / P given in Table IV impose definite limitations on the collision process, and some possible collision types may be examined to ascertain which, if any, give values of p approximating those of Table IV. The following collision types are considered: A. Perfectly elastic-hard sphere collisions. B. Elastic-inelastic-in which the air molecule strikes a portion of the oil molecule elastically and the latter shares its newly acquired transverse momentum with the parent molecule inelastically. C. Inelastic-a perfectly inelastic collision would require that the air molecule remain attached t o the oil molecule after collision. This process appears highly improbable. D. Inelastic-inelastic-this is the inelastic counterpart of B; * it appears highly improbable, as in C. Employing Newton's laws, calculations were made t o determine p / P for collisions of types A and C. For collisions of type B, p / P was set equal to that given in Table IV and a calculation made for m (the mass of that portion of the oil molecule which presumably is first struck by the air molecule, and which acts independently at the instant of impact). Results are shown in Tables V and VI. From Table V it is evident that simple elastic and inelastic types of collisions lead t o values for p / P which are far greater than those observed. An elastic-inelastic type of collision will give reasonable results, however (Table VI), if t h e air molecule collides with one or a t most two hydrogen atoms on the periphery of the oil molecule. No particular significance is attached to the precise numerical values of m beyond the general indication that a single hydrogen atom (or perhaps two hydrogen atoms) would appear to be acting independently of the oil molecule at the moment of impact. It is accordingly suggested that elastic-inelastic
Table IV. Ratio of Acquired to Original Momentum, p / P , after Direct Hit by a n Air Molecule E 0.107 0.119 0.117
Oil Amoil Amoil-S Ootoil
Table V. Oil Amoil Amoil-S Octoil
PIP 0.0278 0.0283 0.0284
Calculated Ratios of Acquired to Original Momentum, p / P , for Direct Hits Experimental, from Table IV
0.0278 0.0283 0.0284
Elastic
Calculated Inelastic
0.480 0.456 0,432
0 240 0.228 0.216
Table VI. Calculated Average Masses of Oil Molecule Fragments which Act Independently on Impact Oil Amoil Amojl-S Octo11
m(Mass Units)
1.58 1.71 1.84
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INDUSTRIAL AND ENGINEERING CHEMISTRY
collisions between the air molecules and small (mass 2 or less) loosely joified segments of the oil molecules account for the small average deflections which were observed. Simple elastic or inelastic collisions between the air molecules and the oil molecules acting as units cannot account for the results obtained by the authors.
Vol. 40, No. 5
A theoretical interpretation of the results suggests that the hydrogen atoms on the periphery of the oil molecules provide springiness to the oil molecule by first absorbing the momentum . acquired a t impact themselves and later sharing this momentum with the balance of the oil molecule. The oil molecules do not act as hard, or as simply deformable spheres, on impact with air molecules.
Summary
Literature Cited
Experimental measurements were made on the deflections of oil molecules in air a t reduced pressures. These indicate that the approximate straight-line path for a n oil molecule is much greater than its mean free path. This is in agreement with the general observation that many high vacuum devices, such as diffusion pumps, operate a t pressures considerably higher than they would if the oil molecules ware appreciably deflected on a single encounter with an air molecule.
(1) Kapff, S. F., and Jacobs, R. B., Rev. Sei. Instruments, 18, 581 (1947).
(2) Kennard, E. H., *“Kinetic Theory of Gases,” pp. 115-34, New York, McGraw-Hill Book Co., 1938. (3) Knauer, F.,2.Physik, 90,559 (1934). (4) Loeb, L. B., “Kinetic Theory of Gases,” p. 98, New York, McGraw-Hill Book Co., 1934. RECEIVED December18 1947
Surface Phenomena Useful in Vacuum Technique Le Roy Apker GENERAL ELECTRIC COMPANY, SCHENECTADY, N. Y.
M
0D ER N v acu u m Studies of surface phenomena can give valuable inforsion pattern. Residual gases techniques can promation at pressures so low as to be unmeasurable by the are adsorbed selectively by duce pressures SO low that usual methods. Previous work o n thermionic emission the various vposed crystal they cannot be detected with from wires, and on field emission from single crystals, is faces, and the pattern a v a i l a b l e gages. Experimentioned briefly. The photoelectric emission from tungchanges radically as the surments in Nottingham’s labosten is very sensitive to residual gas in vacuum systems face becomes contaminated. ratory have shown that the and can be used to estimate partial pressures of active Because the gas condenses on common form of ionization gases. A simpler method involving a sudden burst a relatively cool substrate, of adsorbed gas into an ionization gage is also described. this technique gives informagage fails a t about 1.0-6 miwon. This effect is due t o tion that cannot be obtained x-rays generated by electrons from thermionic emission striking the grid (6). These rays eject photoelectrons from the colmeasurements. The tube walls must be very clean, since the lector and produce the same effect as positive ions flowing to this electrons are accelerated by several kilovolts. Thus, gas can be electrode. Thus the gage indicates micron, even though dislodged by impinging electrons. This field emission method the actual pressure is lower. Obviously, other methods are has been used in a valuable study of techniques for reducing needed for measuring the ultimate vacua attainable today. active gas pressures (4). At 10-5 micron, enough molecules strike a clean surface to Photoelectric Emission from Tungsten cover it with a monolayer in a few minutes. The adsorbed gas may produce radical changes in the behavior of the material. IB the laboratory at Schenectady, a procedure combining EvidentlY, experiments must be done quickly. For this reason, features of both thermionic and field emission methods has been little is known about Clean Surfaces, except for a few refractory used which involves measuring the photoelectric emission from a metals and several materials that are easily evaporated. For tungsten ribbon as a function of time ( 1 ) . The photocurrent this reason, also, adsorption phenornena may be used as sensitive from a metal increases approximately as the square of the quandetectors of certain gases. tity, hv ‘p. Here hv is the energy of the incident photons, and p is the work function of the surface. The primary effect of Thermionic Emission from Tungsten Wires contamination is to change the latter quantity. Therefore, the Langmuir’s experiments on adsorption by tungsten filaments photoelectric method is not so sensitive as the thermionic, which (3) showed that the thermionic emission from these wires Tvas Both techniques involves currents varying as exp (-p/lcT). extraordinarily sensitive to surface contamination-for example, measure an average effect for a polycrystalline surface and cannot cesium increased the current at 800” K. by a factor of 1020 detect highly selective adsorption by preferred crystal faces. At 1500” K., oxygen decreased it by a factor of 106. This deDespite these drawbacks, the photoelectric method is convenient crease is used in one modern form of leak detector (6). because it can be used a t low temperatures and because little or no accelerating voltage is required. ~
-
Field Emission from Single - Crystals
In Nottingham’s laboratory, a different method has proved successful ($1. A phosphor Screen collects the field emission from &verysmall single crystal of tungsten. After the metal is cleaned by flashing at a high temperature, it gives a characteristic emis-
A typical photocell described in (1) was sealed off and was gettered with tungsten vapor. After the tube had been standing for a week at 3000K,, no photocurrent was ejec$ed from the tungsten cathode by radiation of wave length 2537 A. (hv = 4.89 electron volts). The ribbon was then flashed for 10-second periods a t
