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INDUSTRIAL AND ENGINEERING CHEMISTRY
Acknowledgment Acknowledgment is made of the grant of a fellowship and equipment funds by the National Research Corporation, making it possible to carry out the laboratory study of water vapor absorption. Thanks are due both Servel, Inc., and the National Research Corporation for permission t o quote performance data on absorbers.
Nomenclature
cs = concentration of surface solution a t gas-liquid interface D = molecular diffusivity, sq. cm. per second = -c* -
co
- cs R = liquid film thickness, cm. cz
T
=
u = velocity, cm. per second e = distance in direction of diffusion, em. y = distance in direction of fluid flow. cm.
Literature Cited Carrier, W.H., Cherne, R. E., and Grant, W.A., “Modern Air Conditioning, Heating, and Ventilating,” p. 290, New York, Pitman Publivhing Co., 1940. Flosdorf, E, W.,Hull, L. W., and Mudd, S.,J. Immunol., 50, 21-54 (1945).
c = concentration, gram moles per cc. cs = average or bulk concentration of entering solution co = average or bulk concentration of solution leaving
F
Vol. 40, No. 5
DX -, (e taken as total length of wetted surface) uR
Flosdorf, E. W., and hludd, S., Ibid., 34,469-90 (1938). International Critical Tables, Vol. 3, p. 369, New York, McGrawHill Book Co., 1938. McAdams, W.H., “Heat Transmission,” 2nd ed., p. 260, New York, McGraw-Hill Book Co., 1942. Newman, A. B., Trans. Am. Inst. Chem. Eng., 27, 328 (1931). Shackell, L. F., Am. J.Physiol., 24, 325 (1909). Sluder, J. C., Olsen, R. W., and Kenyon, E. M., Food Tech., 1, No. 1, 85 (1947). RECEIVED November 20, 1947
Mechanism of Sintering A. J. Shaler and John Wulff MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MASS.
To
explain the phenomena of shrinkage and expansion which occur during the heat treatment (sintering) of compacts of metal powders the authors conclude that a mechanism exists whereby the metal flows viscously under the influence of surface tension and gas pressure. Using a simplified model, the rates of shrinkage of copper compacts a t 850” C. are calculated and measured. The preliminary measurements confirm the nature of the mechanism but indicate a slightly higher viscosity coefficient than is predicted on the basis of the self-diffusion coefficient of copper.
S
EVERAL investigators in the field of powder metallurgy, among them, Delisle (S), and Libsch, Volterra, and Wulff
(16), have shown that a n uncompressed mass of metal powder
consisting of a single constituent, and heated t o a temperature less than its melting point, can reach in finite time a density in excess of 90% of the theoretical maximum. I t has been shown ( 7 ) that such shrinkage does not require the presence of a liquid phase, although, as in the manufacture of cemented carbides (18), one is often used. Qhrinkage is one of the basic observations which must be explained before the mechanism of sintering can be said to be understood. I t is practical, in the powder metallurgy industry, t o compress the powder first before heating it, and i n that case, another phenomenon appears: that of expansion during subsequent heating. There is some evidence that these two types of volume change stem from different origins. For example, hollow cylinders compressed from both ends are often observed to shrink radially while a t the same tinie they are expanding in an axial direction. Drapeau (15) has made careful experiments on this subject and has shown also that cylindrical compacts may, i n the course of a n extended heat treatment, at first shrink and subsequently expand. One explanation for expansion .and shrinkage, developed by Balshin (1) and reiterated in recent literature ( 9 ) ,is that changes of volume during sintering are due t o selective recrystallization starting at points where, in pressing, selective cold work has taken place. The experiments mentioned above on uncompressed powder sho\V that, shrinkage can take place without cold work.
That expansion is also independent of cold work is shown by the folloM ing experiment: A mass of uncompressed copper of apparent density 4.9 grams per cc. was heated i n argon for 16 hours at 850“ C. Shrinkage caused the density to rise to 7.1 grams per cc. The same compact, by that time unquestionably annealed, was then heated i n vacuo for 8 hours; i t expanded during this treatment to a density of 5.3 grams per cc. The density did not change upon further heating in vacuo for another 8.5 hours. Such evidence suggests that recrystallization as a consequence of cold work takes place in powder no differently than i t does i n solid metal, and that it is not related to the changes in volume of the compact. The central problem of sintering may then be defined as the mechanism whereby powder part,icles adhere t o each other and whereby the mass of them proceeds towards a n equilibrium density. The first of these questions has been discussed a t length, though, until very recently (14),only qualitatively. The reader may consult reviews of the subject (7, 11, f7),which conclude that metallic surfaces in close pioximity are attracted to one another by the electrostatic force field of the surface layers of atoms and their electrons, and that this attraction approaches in magnitude the cohesive force between atoms in the metal lattice. Metallic particles close to one another can therefore form bonds indistinguishable from bridges of solid metal. The serond problem, that of tlle niechanism whereby powder masses shrink or expand in order to attain an equilibrium density, requires a study of the equilibrium structure and of the forces causing the mass of powder to progress towards it. The rate of this progress depends on a third factor-namely, the resist,ance offered by the metal to such forces.
Equilibrium Structure A mass of powder differs from a solid piece of the same metal chiefly in that it has a great deal more surface per unit of mass. Gold powder of 0.1-micron particle size, for instance, has so much surface (over 6 X 1 0 6 square em. per mole) that its free energy is more than 350 cal. per mole greater than that of solid metal (6). One would expect, then, that the transformation of a mass of powder into a piece of different density could be likened
.
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INDUSTRIAL AND ENGINEERING CHEMISTRY
to a chemical reaction, and might be studied thermodynamically. If this is done, i t is soon realized that, because i n thermodynamics a solid does not flow (4, the skeleton formed by the centers of the particles and the bridges between them cannot change. This approach leads to a n equilibrium structure w$ich is not a dense solid, but rather a firm network of metal i n which there are dispersed a great many pores of spherical shape and of volume equal t o that of the voids initially present between the particles. The density of the mass is unchanged. I n other words, thermodynamics can tell us only that the pores eventually become spherical, but nothing about how these pores can shrink or expand with consequent changes in the density of the porous mass. The thermodynamic analysis, therefore, emphasizes that two equilibrium structures exist. One of these is a porous mass of the same volume as the original mass of powder, but in which the voids have become spherical. The change in shape of the pores might be accomplished i n one of three ways: Atoms from one point on the surface of the pore can find their way t o another point by evaporation and subsequent condensation; by diffusion along the surface; or by diffusion through the body of the metal. The other equilibrium structure is a dense body. (It will be shown that entrapped gases can, in certain cases, prevent the attainment of this ideal structure.) The powder mass can proceed toward this equilibrium only if the metal is capable of flow-that is, only if the metal is not a solid i n the thermodynamic sense, but a fluid.
slow compared with the rate of shrinkage of the pores by 'flow under the influence of surface tension, Nevertheless, the rate of shrinkage may be calculated without serious error on the basis of a model in which the agglomerate of powder particles is represented by a mass of metal in which there are distributed spherical pores equal in volume and in number of the voids between the particles i n the mass of powder. The reasonableness of this model (14) is evident when i t is observed t h a t the stress system introduced by the surface tension becomes, a t a short distance from the pore surface, almost indistinguishable from the stress system set up by a spherical void. Mathematically, in the viscous flow of a n incompressible fluid the velocity vector is irrotational, and therefore independent of the shape of the source (10).
Rate af Evaporation and Condensation of Copper Gas at 850" C. From elementary kinetic theory the rate of evaporation of 8 gas from a surface above which the equilibrium pressure is p , , and the rate of condensation on a neighboring surface of pressure pz is G = 21.9 X 10-6 m
(pi
-
pz)
where M is the atomic weight, T the absolute tem erature. The equilibrium pressures are found in terms of the rack of curvature of the surfaces RI and RZfrom the Kelvin equation of thermodynamics I n pi - - 2oM 1 1 pz
Mechanism of Flow Powder and dense metal differ particularly in their surface area; hence i t is clear that the agency causing the flow is the surface tension, defined by Gibbs (6) as the variation of free energy with surface area. Gibbs has shown that the effect of surface tension on a curved surface separating two phases is equivalent t o a pressure perpendicular t o the surface and of magnitude inversely proportional t o the radius of curvature. The surface tension (1200 dynes per cm.) of solid copper accordingly acts on the metal surrounding a spherical pore of 0.1micron radius as would a negative hydrostatic pressure of 3500 pounds per square inch. Such a stress is even sufficient to induce plastic flow by slip, but this type of deformation must be eliminated from present considerations because it occurs almost instantaneously upon the application of the stress. The purpose of this work was to investigate a phenomenon which is time-dependent. Furthermore, the high value of stress can cause little flow in an incompressible substance unless flow also takes place deep in the metal, away from the pore surface. Here the force is spread over a n area increasing as the square of the distance from the pore center, so that the stress (the force per unit area) decreases rapidly. I n fact, in the experiments described, the pore radius of 100 microns is such that at a depth of 0.1 radius beneath the surface of the metal the stress is less than 3 pounds per square inch. Flow at such values of stress has not been extensively measured. The data that are available (9, 18) indicate that the flow is viscous; it has the characteristic that the rate of strain is proportional to the stress, the factor of proportionality being defined as the viscosity coefficient. For stresses up t o over 100 pounds per square inch the viscosity coefficient is constant (or very nearly so) for the metals that have been studied. Kanter (8) and Frenkel (4) have both shown that the existence in metals of the phenomenon of self-diffusion entails the existence of a process of viscous flow in which the moving units are single atoms or 'single lattice defects. Calculations of the rate of sintering of a simple model of a powder compact can be made on the basis of the concepts outlined. It can be shown that in copper, at temperatures near 800 O t o 900 O C., the rate of spheroidization of the pores by surface diffusion or by transfer of atoms through the gas phase is very
T
pRT
(81 -
a)
in which u is the surface tension, p is the density of the solid, and R is the universal gas constant. From thermodynamics, the pressure of an ideal gas is related to the free energy of sublimation AF; by AF; = - R T I n p
The Kelvin e uation may be solved for a n y radius of curvature R, if Rz is ma%e infinite (flat surface), by using the known value of AF; for massive metal. For copper a t 850" C., AF&b, is 46,700 cal. per mole (14), and the corresponding equilibrium gas pressure 1.1 X dyne per square cm. Using the value 1200 dynes per cm. for the surface tension, the rate of evaporation to a region in a pore where the radius of curvature is - 10-2 cm. from a region where the surface is flat is G = 21.9 X 10-
= 21.9
4% c1 -
[exp
( - -3)] -2 X 1200 X 63.6 (8.9 X 1123 X 10-2 X 8.3 X
d0.0566e-m.n X
X
lo?)]
1.84 X 10-6 =
1.05 X 10-19 g./sq. om. sec.
The rate of change of the radius of the curved surface in this case is then
- -atd R
-1.05
x
= -1.2
8.9
x
10-20 om./sec.
As is shown below the rate of change of radius by flow under the influence of the surface tension u is
where 17 is the viscosity coefficient, found experimentally to be 5 X 10+6 seconds per cc. Hence for flow
-5 dt
= -0.75
X 1200 X 0.2 X 1 0 3 = -1.8 X 10-4 cm./sec.
This rate is some 1OI6 times faster than the rate of spheroidization by evaporation and condensation for apore 10-3cm. in radius. Data are lacking for a similar uantitative treatment in the case of spheroidization by surface & € u s i o n ,but the rate is probably not nearly 10'6 times faster than for evaporation. The absence of apparent spheroidization (14) in photomicrographs of Copper sintered at 850' C. supports this conclusion. The rate of surface dR diffusion t o give a value of -- near cm. per second, would dt
.
INDUSTRIAL A N D ENGINEERING CHEMISTRY
840
Vol. 40, No. 5
The energy connected with the flow of the large mass of metal, dR each element of which moves at the rate -, is dt
LOG R /
-'4 -2
This energy is balanced by the energy gained in reducing the area of the pore, or Log F = Log
A'
p2
dt
4-K ( T )
Figure 1. Change of Radius, R , of 4 Pores of Different Sizes in Copper as a Function of the Paramete-r, F . Time, t, temperature, T:and density,
p,
whence
are variable
-4pore initially of radius R, then has a t any later time, t , a ra-
have to be 1Oitimes the known rate ( I S ) of self-diffusion of coppei (4 X gram per square em. second a t 850" C.), an unlikely
dlus
figure.
Let
Rate of Sintering Frenkel (4) has bhown the pattern of the calculations 1eyuii ed t o find the rate of slirinkage of a gpherical void surrounded by a viscous massive metal. The same pattern is followed here, with the exception that the porosity of the metal surrounding the void is taken into account. Othcr variables are also introduced: The presence within the void of a. gas which is indissoluble in the metal; of a gas rrhich forms a compound with the metal; or of a gas which can diffuse through the metal. These more complex oases are treated in greater detail elsewhere (14). I n the figures shown, the rates of shrinkage or expansion are given for powder masses having pores uniformly of the size read on the ordinate (logarithmic scale). The rates are given in terms of the parameter, F , shown in logarithmic scale along the abscissa. The parameter, F , is given by the formula:
where 7 is the viscosity coefficient, which is determined for each metal by the temperature of sintering; its value will be discussed later. The time, t, is the variable affecting F , but p , the density of the compact, also changes n i t h time. The density is introduced into the formula a s a fraction of the theoretical density of the massivr metal. I t s value a t t = 0 is the initial apparent density of the pox-der mass. The surface tension of solid copper has not been measured but there is good theoretical evidence (22) that it has a t all temperatures very nearly the value of 1200 dyne-. per em., as measured on liquid metal a t thr>mdting point. This value wab used in calculating the curves shown.
then
F ~ 5 - g 2u
20
The curves of Figure 1 are lotted from this equation. When there is gas inside the pores txe energy of compressing that gas must be taken into consideration, as is done in the curves of Figures 3 and 4. In Figure 1, the changes in the radius of pores initially 10-1, and 10-4 em. are shown as functions of F , if no gas is trapped within the pores, and when sintering is carried out in vacuo. The method of using such a plot is illustrated by an experiment. An unpressed compact was made up of spherical copper powder particles all nearly of the same size, 0.011 em. The apparent density of the powder mass was determined to be 0.53 times the density of copper. Since the relative density of a simple cubic array would have been 0.52, it was assumed that the number of voids per particle was one. The volume of each pore was therefore 5 x 10-6 cc., and the radius of an equivoluminous sphere approximately 1.08 X 10-2 em.
lo-*,
Calculation of Curves for Figure 1 Consider a spherical pore of radius R in a large mass of porous metal of density p (as a fraction of the solid density). i l t a distance, A , from the center of the pore the volume of material crossing the boundary of a sphere of radius A is the same ( 4 ) as the volume of metal i p = 1) crossing the initial surface of the pore, or
Figure 2. Change in Average Pore Radius, R , in Unpressed Copper Powder Sintered in Argon at 1 Atm. and in Hydrogen at Reduced Pressure (850" C.) Pore size was made a m uniform as possible. The upper abscissa scale and the full line are displaced to thc right by an amount K ( T ) corresponding to a coefficient of viscpsity 5 X 10-6 second per ccs
INDUSTRIAL AND E N G I N E E R I N G C H E M I S T R Y
May 1948
841
sizes is expected to shrink a t first, to expand subsequently, a n d finally, to remain a t an unchanging density. Such a sequence is clearly shown by Drapeau in copper compacts (16). If the compact is cbmpressed before sintering, the pressure of gas within the pores is greater, and the critical radius is less. The pore sizes are also less. .1.0
.
*2.01-+ LOG R
I ATM. IN AND O U T
INS106 ,2.I
Figure 4. Three Cases of Sintering Vacuum sintering with 1 atm. of noble gas inside the pores; vacuum sintering with a reduced pressure inside the pores: sintering with noble gas at 1 atm. inside the poree, and outside the compact. The period of the volume change is the same in all cases
Figure 4 shows three calculated Eurves illustrating the fact that the period of shrinkage or expansion is governed by the pore size only, and not by the gas pressure, which controls the direction and extent of the volume ehange. The three curves are for two cases of vacuum sintering (one with an internal pressure greater than critical and one with a low internal pressure of gas) and for the case of sintering in argon a t 1 atmosphere, with argon at the same pressure inside. -2 -
Effect of Prepressing
LOG R
: -5
I
'9
-8
-7
-6
LOG
P
-4
Figure 3. Change in Radius, R, of 5 Pores of Different Sizes in Copper as a Function of F when a Pressure of 1 Atm. of a Gas Noble to Copper Exists Inside the Compact
until their pressure balances the negative effective pressure which is introduced by the surface tension; after this balance is achieved the pore size remains constant indefinitely. Such a case is illustrated by the curves of Figure 3, in which the values of log F are plotted against the logarithm of the pore radius for compacts sintered in vacuo but having, inside the pores, a noble gas a t an initial pressure of 1 atmosphere. Pores larger than a critical size (2.4 X 10-8 cm. radius) expand to a constant size. Pores of the critical size neither expand nor shrink. Pores finer than the critical size shrink t o a constant size. I n particular, a definite period of time is connected with the change of volume of each size of pore. Thus, fin? pores of radius 10-6 cm. go through their shrinkage period before the pores of 10-2 cm. radius have begun to expand. A compact containing pores of a range of
The analysis presented is applicable to the sintering of masses of uncompacted powder, and perhaps, by extension, to the sintering of pressed compacts of hard powders for which the strain a t yielding is very nearly equal to the strain a t fracture. In such cases the pores are essentially equiaxed and no great departure from reality is made by assuming that they are spherical. The effect of pressure on hard powders is to decrease the pore size by fragmentation; gases trapped within these voids have a goad chance to escape before the pores become isolated, so that essentially the conditions of Figure 1 are approached. Lack of data on the surface tension and on the self-diffusion coefficient make it impossible a t present to make any calculations of F values for tungsten, typical of this class of powders. In the more common case of metals that can be considera)bly deformed, the usual type of unilateral pressing used in powder metallurgy not only decreases the size of the pores, but also raises the pressure of gas trapped between the particles and a t the same time tends to alter the shape of the voids from near sphericity to an ellipsoidal or discoidal form. The analysis outlined, whereby the surface tension is opposed by gas pressure in its tendency towards shrinking the voids in the compact, can be applied to such pores. The gas pressure remains the same everywhere in each pore, but the equivalent pressure introduced by surface tension is greater at the edge of the disk than a t its more nearly plane faces. The inward motion of the pore surface is therefore more' rapid at the edge than at the center. If the gas is highly compressed by the compacting operation the motion a t the faces may be reversed; the faces of the disk might then bulge outward, the
842
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
Balshin, M. Y . ,Vestnilc Metalloprom., 16, 87 (1936). Chalmers, B., Proc. Roy. SOC.( L o n d o n ) , 156, 427 (1936). Delisle, L., Trans. Am. Electrochem.Soc., 85, 171 (1944). Frenkel, J., J . Phys., U.S.S.R., 9, No.5, 385 (1945). 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 (1) (2) (3) (4) (5)
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 mm. 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
