THE KINETICS OF THE SINTERING PROCESS
By
AMOS JOHNSON SHALER
S.B. Massachusetts Institute of Technology
1940
Submitted in Partial Fulfillment of the
Requirements for the Degree of
DOCTOR OF SCIENCE
from the
Massachusetts Institute of Technology
1947
Signature redacted
Signature of Author
Department of Metallu
May, 1947
Signature redacted
Signature of Professor
in Charge of Research
Signature of Chairman
Department Committee
on Graduate Students
/ea
7
V --.
/
Signature redacted
.r,
ACKNOWLEDGMENT
T.o Professor John 'iulff the writer wishes to express his
heartfelt thanks for suggestions and advice concerning the problem and
the experimental work carried out.
For advice and the privilege of personal discussion of certain problems which arose in connection with the thesis, particular
thanks are extended to Professors William P. Allis and Isadore Amdur,
and to Dr. Charles Kittel and Mr. John C. Fisher.
To his student colleagues the writer wishes to express his
deep appreciation for their aid in the discussion of techniques and for
assistance in the preparation of samples.
This opportunity is also taken to thank the various members
of the Staff of the Department of Metallurgy for special attention he
has received from them during his sojourn at the Institute.
7
1
0
SUMMARY
In this thesis a theory is presented which accounts for the
experimental results of the author and of others on the sintering of
metal powders.
Previous attempts to explain the phenomenon of sintering
have been obscured by an improper assessment of the role of such transient effects as recrystallization, desorption and expulsion of gas
from the metal during heat-treatment of the powder compact.
According
to the theory presented sintering is attributed to a viscous flow of
metal under the influence of surface tension modified by a gas pressure.
Thus, in compacts which contain a range of pore sizes, the finer pores
shrink before the larger ones.
If a gas exists under pressure in the pores,
the finer pores shrink and subsequently the larger pores expand until all
pores reach a stable size independent of the temperature.
In the ideal
case where no foreign gas is present complete densification of a powder
compact would occur below the melting point of the metal.
The time re-
quired for such densification is primarily temperature-dependent.
In this picture of sintering it is shown that the force initially
responsible for adhesion between the particles of the compact is the electrostatic field of the electrons in the surface phase between the solid
and its vapor.
The same field modified by the non-electrostatic stress
due to the electron gas pressure, which becomes predominant in the electric double layer of the surface accounts for the surface tension of the
solid metal.
This force in conjunction with the pressure of any gases
present in the pores is primarily responsible for observed changes in
density of the compact.
It induces a flow of a viscous nature.
The
heat of activation Q of the units of flow is measured for copper to be
about 85000 calories per mol and the flow is therefore slower than is
to be expected on the basis of self-diffusion, the heat of activation
of which is 60000 calories per mol.
INDEX
Page
A. Scope and Contribution
C.
Eqilibrium Considerations
D.
The Mechanism of the Reaction
.
The Starting Material
.
. . . . . . . . . . .
. . .
A.
The Force of Adhesion
B.
The Surface Tension
.
.
C.
The Flow
.
.
.
.
.
6
0
~
~
0
0
9
~
34
S
6
0
0
6
S
0
~
0
~
55
S
S
~
~
9
S
~
S
9
0
63
0
*
~
S
5
0
0
0
0
~
85
*
.
.
A.
The Case of Vacuum Sintering
*
B.
The Case of Gas Entrapment
C.
The General Case.
D.*
The Influence of Errors in the Constants for the
.
.
.
0
0
.
.
.
.. . .
.
..
. .
.
.
Metal
. .
.
. .
Calculation
.
.
.
.
.
.
.
.
.
.
.
VIII.
106
.
. .
.
. .
.
. . *
110
.
. .
. . . . . . .
A.
Sintering without Entrapment of Gas
B.
Sintering with Gas Entrapment
C.
Influence of Pore Size .
D.
Influence of Temperature on Heat of Activation
E.
Metallography
.
.
. .
.
.
.
.
.
.
.
. .
.
.
.
. .
.
Discussion of Results. . . . . . . . . . . . .
Conclusion
Bibliography
. . . . . . . ..
....
...
. . . . . . . . . . . ...
Biographical Sketch
Appendices I to VII
. .
.
.
.
.
...
. . . . . . . . . . .
126
.
.
128
.
.
.
118
..........
.
113
129
.
. .
.
.
.
Experimental Results
132
.
.
..
..
. .0
140
.
VII.
. . . . .
.
VI.
105
The Correspondence between Time and Temperature in
Sintering
V.
.
.
.
.
.
.
.
.
91
.
E.
IV.
7
Theory
.
III.
. . .
B.
.
II.
. .
.
Introduction
.
I.
142
150
INDEX OF FIGURES
Figure
Page
1.
The Powder used in the experiments.
.
.
.
2.
Specimen sintered in vacuo 45 hours
.
.
........
.
.
.
.
.
.
.
15
.
3. Specimen heated in argon 16 hours, then in vacuo 24 hours
4.
8
.
.
15
.
Densification of copper compacts compressed at two different
pressures, after Trzebiatowski
.
.
.
.
.
24
6.
Potential Energy Level diagram in the vicinity of surfaces .
40
7.
The Force of adhesion between two surfaces
.
.
.
.
.
.
.
.
50
8.
The Force of attraction between two spheres
.
.
.
.
.
.
.
.
53
9.
A physical picture of the surface tension term due to
electrostatic attraction
Sa.
5.
.
.
. .
.
.
.
.
.
.
.
.
.
.
56
. ........
Two viewpoints in the study of the shrinkage of compacts . .
77
The coalescence of two spheres
.
.
.
.
.
.
79
. . . . . . . . .
.
.
81
.
..
88
. .
90
.
94
.
.
.
.
.
.
.
.
.
10.
The force field on a cylindrical pore
11.
Sintering curves for vacuum case
12.
Variation of the viscosity coefficient with temperature
13.
Sintering curves for case of entrapped gas at 1 dyne/cm2
pressure, vacuum sintering
14.
.
...
.
.
.
.
.
...
.
.
.
.
.
.
.
.
.
.
.
.
.....
.
. .
.
.
..
.
.
95
.
.
.
.
.
.
.
.
96
......
2
Sintering curves for entrapped gas at 109 dynes/cm
pressure, vacuum sintering
...
Sintering curves for compressed copper compacts
18.
Sintering curves for a pressure difference of one
dyne/cm2 with compound formation
.
.
.
.
97
...
......
...
17.
19.
.
Sintering curves for entrapped gas at 106 dynes/cm2
pressure, vacuum sintering
16.
.
Sintering curves for entrapped gas at 1000 dynes/cm2
pressure, vacuum sintering
15.
.
.
Sinter=ng cjrves for a pressure difference of 103
....
..
dynes/cm" with compound formation
.
.
.
.
.
.
.101
.....
..
98
..
.
102
Figure
20.
21.
Page
Sintering cjp'ves for a pressure difference of 106
. . .....
dynescm4 with compond formation
Sintering curves for a pressure difference of 109
dynes/cd
22.
103
with compound formation . . .
.
.
.
.
. .
104
.
Sintering curves for pores 10-C 2 cm in radius under
various conditions
........
.
......
.
107
.
23.
Specimen heated in vacuo 45 hours at 8500C . . . . . . . .
114a
24.
Specimens sintered in vacuo; pore radii vs. time . . . . .
124
25.
Specimens sintered in argon; pore radii vs. time . . . . .
125
26.
Specimens of various initial pore radii sintered at
27.
85000 and at 9000C
..
.
..
.*.*.
127
.....
Specimen sintered in vacuum at 85000 for 45 hours; two
methods of preparation
..
..
..
..
..
..
..
129a
. .
28.
Specimen heated 16 hours in argon, then 24 hours in vacuo
29.
Specimen heated 45 hours in vacuo, showing grain growth
.
.
130
131
THE KINETICS OF THE SINTERING PROCESS
I.
Introduction
A.
Scope and Contribution
Even among metallurgists the term nSintering" has several
meanings.
The meaning in which it is to be used in this study is the
mechanism whereby a mass of metallic or non-metallic powder becomes, at
a temperature less than the melting point, a dense body having properties
approaching those of the massive material prepared in some other way
(casting, deposition, etc.).
Reasons for developing a theoretical explanation of this mechanism, aside from the obvious scientific one, lie in the wealth of conflicting hypotheses described in the literature for the last half century,
and in the desirability felt in the powder metallurgy industry of a
solid engineering basis for predicting without elaborate trial and error
methods the correct operational variables required to give the desired
product.
Processes allied to the sintering of powder consisting of a
single component are:
(1) that in which two or more components are
present, one or more of which may or may not exist in the liquid state
at the temperature of sintering; (2) that of the sintering of a mass of
powder under simultaneous application of pressure and heat; and (3)
the
customary process in which the powder is first of all subjected to a
cold compaction and subsequently to the application of heat.
At the
heart of these processes there lies always the simpler problem of the
mechanism whereby uncompressed powder of but a single component becomes
a dense, massive piece of metal.
The present study is therefore confined
to the investigation of this mechanism, although conclusions drawn from
it are applicable to the other processes as special applications.
For further simplification of the problem the study is confined
to the higher temperature range (in industrial practice this is also the
range commonly used).
By "higher temperature" is meant a temperature in
excess of about 0.55 times the absolute melting point, a figure which,
it will be shown, specifies the state of the surface of the material.
In the case of copper the lower limit is then about 45000.
In the case
of tin, the melting point of which is 23200, the "higher temperatures"
begin at less than room temperature, namely about 500.
Again, there are transient effects which occur in the sintering
process during the heating-up period, the length of which is in general
sufficient to allow the individual grains of powder to come to equilibrium
as far as stress relief and surface activity is concerned.
rated in the next section.
This is elabo-
A sufficient heating-up period is assumed to
exist in the discussions below.
The results are therefore not applicable
in the main to the hot-press process wherein powders are sintered by
simultaneous application of heat and pressure, and which is completed in
general in a few seconds.
It is perhaps necessary to emphasize, however, that the results
found here are fully applicable to the usual sintering process in which
the powders are first pressed in the cold and subsequently fired.
The
study is based on uncompressed powders chiefly in order to keep the phenomena which are attributable to the compression from those due to the
sintering mechanism.
The property of powder compacts with which this
paper is mainly concerned is the density.
Industrial specifications tend
to lay more stress on other physical properties, but it is felt that a
variable density is the only fundamental difference between a powder
compact and a massive metal; other properties follow the same laws in
both forms of the metal, and their relations with density have been the
subject of much experimentation and of many explanations (RhlJol,Tr3,etc.)*.
The first subject dealt with is the nature of the starting
material.
The able work of liAttig and of his collaborators (Hu 14-18)
shows that at the temperatures envisaged (above 45000 for copper) all
physically and chemically absorbed gases and all volatile constituents
of the powder are expelled from the lattice and from the surface within
minutes.
Only solid impurities remain.
In the case of highly compressed
compacts these gases are present in the pores, from which they cannot
escape, in the form of a simple gas in equilibrium with the solid metal
phase.
Further, all cold work stresses- are relieved.
The temperature
of the experiments has been kept so high that all recrystallization phenomena (in the sense generally used in the case of massive metal) take
place during the brief heating-up period.
Secondly, the subject of the free energy of metal powder is
briefly considered, and the sintering process is described as a chemical
reaction, the kinetics of which may be found from reaction-rate theory.
After this the literature is reviewed and older hypotheses and
theories of sintering are critically examined.
Finally the underlying physical bases of the mechanism are
examined, and it
*
is found possible to calculate the force of adhesion
The references are given in this form in the appended bibliography.
-
-4
between metal surfaces.
It is shown that the force of adhesion is the
electronic term in the cohesion of massive metal lattices, and that it
is very nearly invariant with temperature, although the cleanliness of
the surfaces and their impurity content has a considerable effect on it.
This force of adhesion is a sufficient explanation for the various phenomena of cold welding and is the principal factor in the cohesion of
cold-pressed compacts.
The calculation of this force is given in Part II,
Section A.
The second section of Part II deals with surface tension.
It
is shown that the force of adhesion found above is one of the terms responsible for surface tension, so that the two forces must not be distinguished, but must be treated together.
The evidence available in the
literature is presented for the existence and magnitude of this surface
tension, for its variation with change in temperature, and for the influ-
ence of closely adjacent other surfaces on the tension of a surface.
Other forces which might be factors in sintering are examined and found
to be negligible or included in the modern concept of the surface tension.
The theory of Frenkel (Fr 8) regarding the flow of the hot
metal under the influence of the surface tension is studied in detail,
and partial confirmation of it
is found.
Further work needs to be done
on the value of the surface tension of metals and on the coefficient of
self-diffusion of metals before the nature of the flow of hot metal under
small stresses can be unequivocally determined.
Such work is beyond the
scope of this thesis.
On the basis of the foregoing results the course of densification of a compact of copper powder is predicted under various conditions.
-
- 5
Proof is presented that the initial densification and subsequent swelling
of heavily compressed compacts can be explained on the basis of the
theory as being due to the presence of entrapped gas, as suggested by
Trzebiatowski (Tr4).
For the first time an adequate explanation is given
for the fact that sometimes a compact first becomes more dense, and only
subsequently swells to a lesser density.
Trzebiatowski's idea that this
is due to the time factor involved in the decomposition of oxides is
shown to be incorrect for slowly sintered compacts, as is the explanation
advanced by Balshin (Ba28) according to which the swelling is due to
selective recrystallization following selective work-hardening.
Finally a way is pointed out whereby the relationship between
sintering temperature and sintering time may be found.
Heretofore the
powder metallurgy industry has found empirically that a decrease in temperature may be in part offset by a longer time of sintering.
The
relationship is clarified by the present theory.
B.
The Starting Material
G. F. H'ttig (Hu2O) has reported results on the evolution of
gases and volatile impurities during the heating of powders of iron, tin,
nickel, aluminum (Hu22), and copper (Hul6).
From these experiments he
concludes (Hul7) that at a temperature which for all the substances investigated falls at about 0.52 of the absolute melting temperature the
last remains of the volatile constituents are expelled.
Since for copper
this temperature is 4320C, and involves heating for two hours, it follows
from the general theory of reaction rates that at 8500 0, at which the
experiments reported below are done, the time required for the removal
of all volatile constituents is to be reckoned in minutes at the most.
-
- 6
Above 40000 also, H1ttig shows that all surface activity such as physical
and chemical adsorption and superficial atomic rearrangements (he finds
evidence of a two-dimensional surface "recrystallization" taking place
between 0.23 and 0.36 of the absolute melting point) is at an end, and
no further changes take place until the metal is heated to near the melting point (above 8150C for copper).
Unless the compact is unpressed and is sintered in vacuum, in
which case it is expected from these results that no gases are left, the
volatile constituents remain in the pores of the compact; and if the
conditions of pressure and temperature are at some later time favorable,
compounds may be formed and may influence the course of sintering.
For
these reasons, the calculations of the progress of densification in copper
during sintering include (Part III):
first, the problem of a compact,
unpressed, heated entirely in vacuo; second, that of an unpressed compact
sintered first in argon, to entrap a neutral gas, and subsequently in
vacuo, to simulate the case of the entrapment of inactive volatile matter;
and third, that of a compact pressed at about 7j tons per square inch in
argon, to entrap considerable gas, and sintered in vacuo.
The same cases,
with sintering done in argon at one atmosphere, and that of a compact
pressed in oxygen at 7j- tons per square inch, to entrap a gas that forms
a compound with the metal, are discussed in detail.
The case of a metal
to which there is added a hydride such as titanium hydride, to provide
hydrogen which fills the pores, is one of considerable interest but also
of considerable difficulty, because the diffusion of hydrogen through
the metal leads to a variable distribution of pressure throughout the
compact.
-
- 7
The temperature selected for the experiments and calculations
is higher than that (72000) at which Sauerwald (Sa2O) (Sal8) (Sa2l) (Sa23)
(Sa27) finds the inception of grain growth (see Part I, Section D) so
that it
is certain that no cold work stress effects remain (Balshin,Ba28).
The powder used is atomized electrolytic copper from McAleer Manufacturing
Co., Rochester, Michigan, and is preliminarily sieved for one hour; only
the size fraction between 100 and 140 mesh is used.
A photomicrograph,
Figure 1, shows that the material is very nearly of spherical shape.
has been subjected to very little cold work.
It
The powder used is all re-
duced in purified hydrogen (dehydrated and deoxygenated) for two hours
at 450 0C before using.
In summary, the evidence of Httig and of Sauerwald indicates
that at 85000, the temperature at which the calculations and experiments
are made, copper powder is free of any surface impurity except oxides
(in the experiments these are removed by treating with hydrogen); and
will not undergo any important recrystallization phenomena.
are compressed and sintered in controlled atmospheres.
The compacts
The starting
material may therefore be described as well known.
C.
Equilibrium Considerations
There is no doubt that a conglomerate of finely divided par-
ticles is in a state of higher free energy than a single crystal of the
same metal.
Unless there is a state of even higher free energy lying
in between these two states, which is doubtful as will be shown below,
then the finely divided powder must spontaneously rearrange itself in
the direction of the more stable state.
There is some doubt that the ideal single crystal is the state
-8-
Figure (1). The powder used in the experiments, viewed at a
magnification of 1001. The top picture shows the powder as it is
after heating in hydrogen at 4500C for two hours. The lower picture
shows the same powder, set in sodium silicate, polished, and etched
with 50% NH40H-50%H202.
-
-9
of lowest free energy at any temperature above absolute zero.
Seitz (Se7)
suggests that in the equation for the free energy
F = H - TS
whereas the enthalpy H is undoubtedly lowest for the ideal single crystal,
yet the free energy is even lower at any finite value of T, the absolute
temperature, if the entropy has a positive value.
measures the imperfection of the crystal.
Now this entropy S
Without entering into the con-
troversy over whether this imperfection is in the nature of lattice holes,
or mosaic or block structures, or dislocations, it is clear that the difference in free energy between the structure of lowest free energy and
the ideal crystal is small and can be neglected in comparison with the
relatively much greater surface free energy of a finely divided powder.
Indeed, Seitz shows that the free energy is least when the number, n, of
lattice defects among N lattice points in a crystal is given by
n/N = e
.. A AbsL.
A LT ; the difference in free energy between the equilibrium
structure containing n defects and the ideal crystal is, however, at room
temperature, essentially zero for copper.
(See Appendix I for this
calculation.)
The free energy of comminuted particles has been estimated by
H'uttig (Hul7) for gold powder made up of cubic particles of various side
length B.
The same calculation is made here for copper powder of spher-
ical shape and of radius n.
Appendix II.
The details of the computation are shown in
The values of AF are
r (cm)_
1
10-1
10- 2
10-3
10-4
10- 5
10- 6
10-7
gas
-
-10
F (cal)
0.000206
0.01177
0.1274
1.284
12.85
128.5
1285.
12850.
91145.
Table I. Differences in
Free Energy between
Comminuted Metal (spheres)
and the Solid Crystal,
per mol, for copper.
given as the difference per mol between the powder and the solid crystal
or equilibrium crystal.
The value for the gas is estimated on the same
basis, that is, the volume of the single atoms is taken as
of the volume of solid metal, per mol.
1.023x10 0
The comparable value of LF'
calculated from the data given by Kelley (Kell) is, for the gas,
A F0 = A H
- TASO = 81525 - 298 x 31.83 = 72035 cal.
The value used for the surface energy is 2535 ergs/cm2 , given by Fricke
(Frl7) for the surface of least energy of the crystal (the (111) plane).
In passing, it might be mentioned that in the various experiments made by Httig and his collaborators, and mentioned above, powdered
copper was used which is specified by the sieve analysis
200-250 mesh,
0-0.75%
250-310 mesh,
0.50-1.75%
less than 310,
remainder
In a very simple measurement (Hul8) of the difference of electromotive force between a heavy copper wire and the copper powder in
question, made both at 2500 and 4000, he was able to show a free energy
difference of about 2000 cal/mol, a value which places his powder in the
neighborhood of the 10-6 cm. range of radii.
Clearly, for the purposes
of sintering kinetics, a specification of powder sizes by their free
-
- 11i
energy values would be much more indicative than the specification by
sieve analysis.
The sieve with 310 meshes per inch passes particles of
radius approximately 2 x 10
3
cm.
The Cenco photelometer (Cel) can
measure particles of radius 5 x 10- 6 cm.
The powder used in the experiments reported here is too coarse
(100-140 mesh) to give any measurement of electromotive force, and was
selected for other reasons.
In summary, it
is possible to both measure and, to a consider-
able degree of accuracy, calculate the departure of powdered materials
from their equilibrium state.
If the mechanism of the reaction leading
from the powder to the crystal is known, it
is then possible by the theory
of reaction rates (Gl) to obtain the rate of sintering.
it
Unfortunately
is not easy to specify the mechanism of the transformation in the
required manner.
Gibbs (G13) has shown (and Defay (De22) has elaborated on the
Gibbsian concept) that the surface between two phases may be treated as
a third phase, the thermodynamic properties of which are, of course,
fully specified by those of the other two phases, to satisfy the phase
rule.
The reaction of sintering is then that between a surface phase,
a gas phase, and a solid phase, all containing one and the same component
only.
The first law of thermodynamics states that, if w is the area of
the surface of separation between the solid and the gas,
dE
= dQ - PIdVI - PtW"
+ (T dw
where T is the surface tension (G13), P' and P" the pressures in the
solid and gas (the gas, in a one-component system, is the vapor of the
metal in equilibrium with the solid metal); the other symbols have their
-
- 12
usual thermodynamic significance.
dQ = TdS
The second law states that
-
dQ'
in which dQ' is written in terms of the various thermodynamic potentials
rlas
dQ'= -dr))dm
dm a
-
The primes, seconds, and ""
superscripts refer to the solid, gas and
surface phases, the its refer to the components.
The mass ma of component
i
1 present in the surface phase is defined, if mi is the total mass of
component i, by
M, - (M' + m' ) = Ma
The dQ and dQt terms may now be eliminated, and the free energy may be
written to include the three phases
dF = -SdT - PtdV1 - P"dv" +c'dw + fj(rt)dm+
(r?)%d
In the case of one component the summations drop out, since there is
only one i.
This leads to a formal definition of the surface tension,
since, taking the partial derivative with respect to w, we obtain
SF/Sw =
The thermodynamic potential coefficients r' are functions of the chemical
potentials usually given by the symbol tk, and also of 'lateral potentials' which express the influence of the concentrations of component i
in the solid and gas on the partial free energy of the surface.
Now the condition which, in the presence of a surface, takes
the place of the condition of equality of pressure between two phases is,
if R
and R2 are the principal radii of curvature of the surface, and R
the radius of total curvature,
-
- 13
= -(Al + /2-
+ 2 q/R
We shall return to the definition of the principal radius of curvature
below (Part II, C).
This is the condition of equilibrium towards which
the reaction tends for which the dF is written above, namely the transfer or material between gas and solid via the surface phase.
Now 4' is
specified by the temperature and the concentrations in the three phases,
and, according to the phase rule, it cannot therefore also be a function
of R.
Therefore if
(
is constant all over a surface in mechanical
equilibrium, R must also be a constant, i.e. the surface must be either
plane or spherical.
In other words, the pores inside a compact of metal
powder must tend, not to disappear, but to become spherical, in the reaction discussed above.
Since the process of sintering includes the complete disappearance of pores from the compact, it is evident that the thermodynamic
approach cannot give us a description of the required reaction.
reason for this is not difficult to find.
it possesses rigidity.
The
For Gibbs a solid is a solid;
It does not flow nor become altered by diffusion.
Its shape can only be changed by evaporation and condensation, or solution
and precipitation.
The concept of affinity of a reaction, and the whole structure
of rate theory built on it, can therefore not be applied here except to
describe the rate at which pores become spheroidized by evaporation and
condensation without changing volume, i.e. without contributing to the
densification of the compact.
To be sure, as long as the pores remain
open and communicate with the outside, there is a transfer by evaporation
from the outside surfaces of the compact and condensation on the internal
-
- 14
surfaces which are in equilibrium with a vapor at a lower pressure.
But clearly this must be a slow process by virtue of the small surface
from which the evaporation takes place (P14). Wulrf shows (Rhl, discussion) that marks made on the surface of a compact do not disappear during
sintering.
The result obtained above gives us a further pointer in the
study of the mechanism of sintering.
It is this:
there are two distinct
processes going on, one, the spheroidization of the pores, contributes
nothing to densification, but eats up some of the surface energy which,
as we shall find, is the driving force of the second process, the reduction in volume of the pores.
Since the first process reduces the free
energy available for the second, the rate of change of density of the com-
pact is thereby lessened, and anything favoring the first opposes the
second.
The calculations made in Part III will show:
(1) that the pres-
ence of a gas pressure (a gas other than the vapor of the metal) inside
the pores opposes the densification process, and (2) that therefore spherical pores are encountered in compacts sintered in gas whereas they do
not occur in compacts pressed in vacuo (or unpressed) and sintered in
vacuo.
Results found experimentally verify this conclusion, as is shown
in Part IV and in Figures 2 and 3.
This much, then, is learned from an A priori study of the Gibbs
treatment of surfaces.
The nature and rate of the second process, that
of densification, must be found elsewhere.
D.
The Mechanism of the Reaction
Jones (Jol) ably reviews the first theories of sintering.
He
disagrees with Endell (Enl) who requires the presence of a small quantity
* -
-
-
- 15
Figure (2). Specimen heated for 45 hours in vacuo, showing no
spheroidal pores. Magnification 5001. 50%H4MH-50%H202 etch.
Figure (3). Specimen heated for 16 hours in argon and then for
24 hours in vacuo. Note tendency of pores to become spherical.
Magnification 1001. 50140H-50%H202 eteh.
of liquid to permit sintering between particles of a solid.
A lique-
faction due to pressures is, for most solids (ice is a notable exception)
precluded by the Clausius-Clapeyron equation, which shows that pressure
favors the phase of lower specific volume.
A liquefaction due to the
fact that small particles have higher surface energy and therefore might
have a lower melting point is shown to be out of the question by Meyer
and Eilender (Me7), who give as the reduction of melting point with decrease
in radius
T-Tr
T0
2 V
r q
sg
lg
where T is the melting point for particles of radius r for a substance
of molecular volume Vm and heat of fusion per mol q.
The T 's are the
surface energies for the solid-gas and the liquid-gas interfaces.
The
lowering of the melting point for particles of radius l07 cm is about
Finer particles can hardly exist without being called gas; and
10000.
all metals sinter at less than 1000C under their melting point.
'
Smith (Sm7) puts forward the theory that sintering can only
take place when a substance changes from one crystalline form to another.
Huttig' s evidence that between 4500C and 8500C no such changes take place
even on the surface of copper is in contradiction with Smith's idea.
Jones himself (Jol) brings out the important point that "the
conditions which give rise to the sintering of a metal powder are identical with those which condition the cold welding of massive materials."
With modifications as pointed out in Part II, his statement that "the
actual forces which effect sintering are the normal effective cohesive
forces within the metal .... and normally decrease with temperature" is
true and acceptable to most modern investigators.
His emphasis is,
17
-
-
however, misplaced, as i' brought out below, because the terms in the
cohesive force involved in drawing metal surfaces together (formation of
the initial bond) are the ones which decrease with increasing temperature.
Once the surfaces are in contact the other terms (non-electrostatic terms
of Samoilovich (Sa9)) are more important than those of electrostatic origin,
and these do not vary appreciably with temperature (the temperature coefficient of the surface tension is very small).
Jones and H~uttig (Jol and Hul7) both discuss in detail the various
measurements, hypotheses,. and experimental methods that have been developed
to determine the temperature variously known as "sintering temperature"
or "temperature of initiation of sintering" or "temperature of onset of
structural alterations."
Jones concludes that "strictly, there is no such
temperature and the expression has no fundamental significance.
tice, it
In prac-
amounts to a temperature at which some change occurs associated
with sintering as recorded by the particular method of measurement."
In
Part II, C, it will be shown that the nature of the flow of the metal
under the influence of the sintering force (surface tension) is more closely
allied to a diffusion process than to a process such as slip or the movement of dislocations which require a definite minimum effort to overcome
a free energy barrier.
Therefore, sintering flow can take place under the
influence of an infinitesimal force.
Since the surface force exists at
any temperature and the diffusion rate is finite at any temperature above
absolute zero, it follows that in accordance with Jones there is no lower
limit except absolute zero to the temperature of sintering.
At low tempera-
tures it is the slow rate of diffusion which prevents most metal powders
from sintering in a reasonable length of time.
But there are factors which interfere with the smooth flow of
-
- 18
the metal under the influence of the sintering force, although in general
a compact heated for a long time at any temperature follows the same course
of changes.
At low temperatures the transient interfering factors are
in evidence for a long time before the normal course of events begins.
At higher temperatures they are in evidence only for a short time, and
for usual sintering temperatures the transient effects are completed dur-
ing the heating-up period.
These effects which include the removal of
adsorbed and chemisorbed gas layers and the relief of cold work stresses,
are at the core of the confusion which exists in the literature on sintering.
Perhaps Htttig's work clarifies the subject better than any other
investigation (Hul7, Hul8).
He finds after reviewing an exhaustive mass
of experimental work by others and by himself that six stages may be distinguished in the process of heating up a powder compact of any metal.
a.
At temperatures lower than 0.23 of the absolute melting
point (4000 for copper) the cohesion of the compact increases continuously
but without appreciable shrinkage.
There is some evolution of adsorbed
gases due to the beginning of a two-dimensional "recrystallization" of
the surface.
There is a marked reduction in the surface area as measured
by adsorbtion experiments.
b.
Heating to between 0.23 and 0.36 of the absolute melting
point (40-2150C foi- copper) leads to a definite swelling of the finest
pores and capillaries.
Hittig ascribes this to a loosening of the surface
due to surface diffusion of atoms.
c.
Between 0.33 and 0.45 (225-3350C for copper) the surface
"recrystallization" is completed.
appear by surface diffusion.
The fine capillaries shrink and dis-
Shrinkage of the compact sets in.
Auttig
19
-
-
states that 'this shrinkage is due to surface diffusion.
must be contradicted.
Such a statement
As has been shown in the discussion of the Gibbsian
treatment of the sintering reaction, a process whereby surface atoms are
evaporated and deposited elsewhere or, and this is equivalent from an
equilibrium if not from a rate point of view, a process whereby surface
atoms move along the surface from one point to another, can lead to sphe-
roidization of pores but not to shrinkage or swelling of the compact.
The reduction of pore volume can only be accomplished by a volume flow
in the lattice of the metal.
d.
This question is elaborated in Part II, C.
From 0.37 to 0.53 (230-4450C for copper) internal lattice
diffusion becomes predominant and destroys the order established on the
surface by the superficial "recrystallization."
established on the surface.
A new grouping is thus
Chemisorbed gases are removed in this range
of temperatures, so that the surface area and surface tension are modified.
e.
From 0.48 to 0.8 and higher (81500 for copper) the lattice
recrystallization is over (this is Hittig's second period of activation)
and the state of the surface does not change again until near the melting
point.
creased.
Shrinkage continues at a faster rate as the temperature is inHlttig ascribes to lattice diffusion the role of "assisting"
in the shrinkage process.
It is shown in this thesis that the diffusion
of atoms through the lattice is the only method of causing the disappearance of spherical pores, and is therefore the only method that can reduce
the volume of the compact.
f.
melting.
Above 0.8 there is a renewed activation in preparation for
HAttig does not go into detail concerning this stage.
At the
highest temperatures new crystal nuclei form and recrystallization
-
- 20
commences; any falling off of mechanical properties may be explained by
excessive grain growth, in accordance with the earlier results of Sauerwald
(Sal8, Sa2O, Sa2l, Sa23, Sa27) and Trzebiatowski (Tr4).
Huttig's review and his experiments in general show that surface
activity of one sort or another exists up to a temperature which is about
450*C in copper; above that temperature, the surface is clean of adsorbed
atoms, and does not change again until the melting point.
This evidence
supports the result of Samoilovich (Sa9) showing that the surface tension
of the solid metal is the same as that of the liquid metal.
Httig's
work further clarifies one puzzling result obtained by Tammann and Mansuri
(Tal2) who found that what they called the temperature of sintering was
the same within 300
for all metals, and lies between 1260C and 15500.
The experiment which led to this result is the following:
stirred by a paddle, and slowly heated.
the powder is
At a certain temperature the
stirrer, which is actuated by a friction drive, stops.
Clearly this effect
is explained by the removal of the physically adsorbed gas which Hittig
finds taking place at 40-2150 C; the removal of this gas permits contacting surfaces to come closer and adhere more strongly (the force of attrac-
tion between surfaces is shown in Part II to increase rapidly as the surfaces approach one another).
It is probable that most of the results
obtained by Smith (Sm7), Schlecht, Schubardt, and Duftschmid (Sc32),
Hedvall (He18), Trzebiatowski (Tr3), and others, giving various figures
for the "temperature of onset of sintering" can be explained satisfactorily
by Hutttig's coirlusions concerning removal of adsorbed gas and a lowtemperature surface "recrystallization"' or reordering followed by a
higher-temperature lattice recrystallization.
21
-
-
Balshin (Ba28) has a totally different conception of the mechanism of sintering.
It
is based on a loose interpretation of the phenomena
of recrystallization; the same looseness is in evidence in some of the
work of Sauerwald.
Balshin states that the recrystallization taking place
in massive metals as a result of cold working and subsequent heating is
a process in which one grain grows at the expense of another.
In a pow-
der compact, he continues, a grain is not bounded everywhere by metal,
and consequently it can recrystallize at the expense of space.
Therefore,
according to Balshin, recrystallization leads to shrinkage of the compact.
That this is a fallacy is easily shown.
Modern theory of recrystalliza-
tion indicates that it is a process in which the boundary between two
dissimilarly oriented grains moves by virtue of the fact that atoms at the
surface of one grain cross the boundary to become part of the other grain.
There is no movement of atoms distant from the boundary from one grain
to the other.
Therefore, if the boundary moves until it reaches an ex-
ternal surface, such as that of a pore, it then must cease to move, because
there are no more atoms which can cross the boundary.
Recrystallization
can therefore not in the slightest degree alter the shape of a metallic
mass, whether it be porous or not.
Experimental evidence of this fact is
found in Smithells' (Sm3) photomicrographs showing the recrystallization
of a "coiled coil" type of tungsten filament.
The resultant single crystal
extends throughout the filament, its crystallographic directions being
entirely independent of the complicated contour of the wire.
has, of course, neither shrunk nor expanded.
The filament
The conclusion is that re-
crystallization has nothing to do with sintering.
If cold work exists in
metal powders, and these powders are heated, recrystallization will go on
just the same whether the powder is all in a heap or compressed or whether
the individual particles are held a mile apart.
Balshin's extension of
his concept to explain the swelling which sometimes takes the place of
the shrinkage in powder compacts on the basis of recrystallization therefore loses its value.
Additional evidence for this view lies in the fol-
lowing observations:
1.
very little
Unpressed powders, which presumably have been subject to
or no cold work, can swell.
(See experiment in Part IV in
which unpressed powder is sintered in argon for 16 hours.)
2.
In some cases, such as in cylinders compressed axially
(Rhl,disc.) shrinkage in one direction can occur simultaneously with
swelling in another direction, or (Tr4) the swelling may be preceded by
shrinkage, so that presumably recrystallization is over before swelling
begins.
Sauerwald's extensive work on recrystallization in powder compacts must therefore be looked at from a new point of view.
To begin with
it must be noted that the word which in discussions of his papers has been
widely translated as "recrystallization" is the German "Kornwachstum" or
sometimes "Kornvergrosserung"
(Sal8).
Sauerwald expresses some surprise
at the discovery of a temperature (72000 for copper) at which apparently
all of a sudden the grains begin to outgrow the initial particle boundaries.
It must be concluded that the phenomenon observed by Sauerwald is merely
the normal grain growth which occurs in unstrained metals at a high temperature.
This phenomenon takes place by lattice diffusion under the
driving force of the intergranular surface free energy, and therefore
should become rapid enough to be observable in a reasonable time at the
23
-
-
temperature at which sintering also takes place at a reasonable rate.
Hence, the confusion in Sauerwald's early paper (Sa2O) between the temperature at which "Kornwachstum" first appears and that at which sintering becomes rapid.
Sauerwald also makes a false statement in an early
paper concerning the dependence of the force of adhesion between surfaces
on temperature (Jol, p. 56), but in later papers he recognizes that (Sa24)
the adhesion force is a part of the normal cohesion in metallic lattices
and therefore decreases slightly with increased temperature.
In general
Sauerwald's investigations in the field of powder metallurgy were centered
on the physical properties of the sintered metal, and not on the process
itself.
On the preceding page mention was made of the swelling of compacts, in connection with Balshint s recrystallization ideas.
The experi-
mental fact of swelling has been known by all powder metallurgists, but
it was Trzebiatowski's experiment which started the polemic on the subject.
In an investigation (Tr4) on copper powder compacts compressed at two
different pressures (42 tons/square inch and 210 tons/square inch) he found
that, in the course of heating, the compacts pressed at the lower pressure
shrank at all temperatures, whereas the compacts compressed at the highest
temperature swelled at all temperatures but the lowest.
He attributed
this swelling to the pressure of gases coming out of the metal where
they were previously held in solution or as compounds.
Their expulsion
is due, according to him, to the change in equilibrium conditions as the
temperature is raised.
In the heavily compressed compact the communicat-
ing pores between particles are very small and the gas cannot diffuse
out as fast as it is formed except at the very lowest temperature that he
studied (10000).
the outside.
24
-
-
At lower compressions the gases have freer access to
In Parts II, C, and III it will be shown that Trzebiatowski's
curves of density vs. temperature of sintering for one hour can be replaced by curves of density vs. time of sintering at one temperature without altering the shape of the curves, although the abscissa scale will
change.
If this is so, then it
is further shown that Trzebiatowski's
interpretation is correct and that the decrease in slope of the low compression curve at the low temperature end and that of the high compression
curve at the high temperature end may also be explained on the basis of
the pressure of the entrapped gas.
Trzebiatowski's curves are reproduced
in Figure 4.
Figure 4, on the next page, shows the density of compacts
compressed at 42 tons per square inch and 210 tons per
square inch.
His X-ray diffraction studies on these copper compacts show that at 40000
one hour of sintering is sufficient to remove all cold work stress.
Shortly after the experiments of Trzebiatowski, Balshin (Ba28)
wrote his principal paper on sintering.
In it he quotes Sauerwald as
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-
- 25
insisting that recrystallization takes place above 72000 and Trzebiatowski
as refuting this statement.
As has been pointed out above, Sauerwald
was talking about a grain growth phenomenon which is not related to cold
work, so that Balshin's criticism is baseless.
Balshin then explains the
swelling of powder compacts by a process of "selective recrystallization"
at points which have received maximum work hardening during pressing,
i.e. the areas of contact.
If the powder particles are large and have a
small specific surface there is also considerable opportunity for centers
of recrystallization to form away from the contact points.
says, favor the "breaking of bonds" and swelling.
These, he
In finer powders there
are many more contact areas and the recrystallization takes place predominantly at these points; in the sintering of fine powders, therefore,
according to him, shrinkage is the rule rather than swelling.
It is
shown in the experiments described in Part IV below that very large
(100-140 mesh) copper powder of very small specific surface (spheres) and
unpressed (very little,
if any, cold work) may be made to shrink if sin-
tered in vacuum, or to swell if
sintered first in argon, then in vacuum.
Such an experiment renders implausible Balshin' s mechanism, and gives
strong support to Trzebiatowski's interpretation.
As Rhines shows (Rhl)
in his able review of the literature on sintering, modern thought supports
the gas-entrapment hypothesis.
Balshin does, however, bring out some interesting experimental
results.
He shows that, in the same die, compacts of different weights
compressed with the same pressure give different green densities (for the
mathematical interpretation of this phenomenon see Shaler (Sh2) ).
sintering, some of these compacts shrink; others swell.
Upon
But if they are
-
- 26
compressed to the same density they follow the same course of densification on sintering.
Such evidence indicates that cold work has little to
do with the course of sintering, and it
is safe to assume as is done in
this thesis that in the case of unpressed powders sintered at 85000 re-
crystallization phenomena do not enter into consideration.
It is further
evident that density is a more fundamental property as far as sintering
is concerned than the compaction pressure, the state of cold work of the
powder, or the mechanical properties of the green compact, although, of
course, all these may influence the green density which is practically
attainable in a powder metallurgical plant.
The foregoing review of the literature on sintering follows
much the same course as does the review by Rhines (Rhl), who concludes
that:
A.
"The initial bond that appears spontaneously at points of
metal-to-metal contact at room temperature is identical in kind with the
forces that hold the atoms of a metal crystal in place and give solid
metals their strength."
B.
"Grain growth will be limited essentially to the powder
particle size until sintering has progressed so far as to provide substantial bridging between particles, whereupon larger grains may grow.
As a result, the temperature of rapid grain growth will appear to be
coincident with the temperature of rapid sintering."
C.
"Recovery, recrystallization and grain growth may be re-
garded as proceeding in a normal manner if due allowance is made for the
influence of special geometrical factors peculiar to powders."
D.
"The temperature of rapid grain growth should not be responsive
27
-
-
to the effects of cold work that are destroyed at temperatures well below
that of rapid sintering."
"At elevated temperatures the movement of metal is presumed
E.
to be accomplished through the action of plastic flow, or of surface
diffusion, or of both acting cooperatively under the influence of the energy
of surface tension as the major driving force."
"Since the proposed mechanism of sintering is, by itself,
F.
capable of predicting no volume changes except shrinkage, growth (swelling)
must be explained by some other process.
The major cause of growth lies
in the expansion of the void spaces in the compact through the action of
gas pressure."
These quotations from the Rhines paper summarize the discussion
made above.
Rhines is not sure, however, whether the sintering process
(shrinkage of the compact) takes place by surface diffusion or by plastic
flow.
It is the major purpose of this thesis to show that surface diffu-
sion can account for the spheroidization of pores, but not for shrinkage
of the compact; that the mechanism of flow is of a lattice diffusion nature,
and cannot be a plastic flow of the type generally thought of in speaking
of massive metals, that is, a shear along particular lattice orientations.
As will be shown in Part II, C, the force field around a spherical pore
due to surface tension cannot be resolved into a system of shear stresses.
It is a pure tension, and due to incompressibility of the metal a spherical
pore is in equilibrium as far as a shear type of flow is concerned.
Therefore, a type of flow must be found which can take place
under the influence of surface tension.
in the literature.
Two suggestions have been made
One is quite recent and was brought to the attention
28
-
-
of the author and translated for him by Dr. G. Kuczinsky, to whom his
thanks are due.
It is in a paper by B. Ya. Pines (Pi4).
occurs by the mechanism of diffusion," he states, "......
iSintering
the greatest role
in this phenomenon is played by the outward motion of voids from the
body.
These voids cannot move out in the form of macroscopic holes (like,
for example, bubbles of gas moving out of a liquid) ."
He then states
that atoms on the surface of the void near the contact points of the
grain have a lower potential energy than the atoms inside the lattice.
These have a high probability of moving into the pore.
This process
creates unoccupied points of the lattice inside the grain near the sur-
face of the pore:
the unoccupied points he calls "holes."
then proceed to diffuse away from their birthplace.
The holes
Some come out again
at the pore surface away from the point of contact, and these do not return again into the metal, for the appearance of a hole on the surface
means a decrement in potential energy.
spheroidization of the pores.
Pines explains in this way the
It remains to be seen whether this process
is more rapid than the surface diffusion or the evaporation-and-condensation mechanism.
In Part II, C, it
is shown that spheroidization of pores
is more rapidly accomplished by the transfer of material through the gas
phase.
Pines' main contribution consists, however, in his explanation of
the shrinkage of pores, whether spherical or not, by the diffusion of holes
through the metal towards the outside of the compact.
He correctly recog-
nizes that "the process shown above (spheroidization) does not mean the
sintering, in the true sense of the word, because it does not show how
the volume-porosity of the body decreases."
To calculate the motion of
holes through the lattice he considers the holes as a soluble substance
-
- 29
being transferred from the voids to the metal, and diffusing through the
metal.
The "osmotic pressuren determining the equilibrium concentrations
of holes in the voids and in the metal is the equivalent inward pressure
due to surface tension, shown by Gibbs to be 2~/R, as discussed above
in Part I, C.
He finds that the equilibrium concentration of holes in
the metal is higher, the shorter the radius of curvature of the surface.
Hence, there must be a net diffusion of holes from the pores towards the
outside of the compact, i.e. shrinkage of the compact.
Applying the law
of diffusion
DVC =
he then finds the rate
C/St
of shrinkage of the compact in the case of a
single spherical pore at the center of the compact.
He then modifies
the expression to include the effect of a very great number of pores
( = sources of holes).
He finds two approximate solutions, one of which
predicts uniform shrinkage throughout the compact, and is true as long as
the linear dimensions of the compact are much less than FD
is the diffusion coefficient of the holes through the metal.
,
where D
The con-
stant 4, is determined from the relation
Q = 'F (Cl - C)
which, by analogy with Newton's heat theorem, replaces the sources (pores)
by a rate term in the diffusion equation.
ckis then a coefficient of
transmission akin to the coefficient of heat transmission in heat theory.
The second solution is true when the dimensions of the compact are larger
than Zj , and predicts a rate of sintering inversely proportional to the
size of the compact.
This solution requires that the outside of the com-
pact sinter first, and as time increases the rate changes to one inversely
-
- 30
proportional to the square of the dimension of the compact.
In all three
cases the rate is also proportional to the concentration of holes near
the surface of the pores.
As the temperature increases the concentra-
tion increases and so does the coefficient of transmission, the former
exponentially.
The larger the compact the more pronounced is the non-
uniformity of sintering; the higher the temperature, the more pronounced
it is.
Pines gives no experimental evidence that compacts sinter from
the outside inward, or reach complete densification in an outer crust
before the center is densified.
The largest compacts mentioned in the
literature are those made by Offermann, Buchholtz and Schulz (Ofl) who
sintered unpressed 2-ton ingots of carbonyl iron powders.
mention of any crust or unsintered center.
They make no
In the experiments described
in Part IV there is no evidence of any such non-uniformity.
Any non-
uniformity, in fact, tends towards the opposite (a sintered center surrounded by a less well-sintered crust), but this is explainable on the
basis of an imperfect vacuum, leaving enough oxygen around the compact
to reoxidize the outer zones slightly during the heat (the compacts were
thoroughly deoxidized in purified hydrogen at the start of the heat).
Consequently it is to be concluded that Pines' last two solutions do not
occur in the usual practice.
His first solution leads to the same result
as those found by the development of Frenkel's viscous flow theory, which
is preferred by the author.
Unfortunately Pines' first solution gives no
inkling as to how the parameter c, is to be estimated.
Nor does he give
any method whereby the coefficient of diffusion of his holes is to be
found, although it is probable that it is simply related to the coefficient
of self-diffusion of the metal.
So it is not possible to decide with
-
- 31
finality whether his first solution is right or wrong.
The second suggestion made in the literature as to the nature
of the flow is that of Frenkel (Fr8). Viscous flow in fluids, he says,
is due to the diffusion of minute cavities of a few Angstr"ms dimension,
due to thermal motion.
In solids they are more definite in size, as they
If U is the energy of activation of a vacant
are vacant lattice sites.
site, then the ratio of the number of vacant sites to the total number of
sites is
U
'IN
= eE
If D is the coefficient of self-diffusion in the metal, the coefficient
of diffusion of the lattice cavities is
DI
If
DN/N'
is the lattice constant, T
the mean life of a cavity at a given
site, y Jo the mean vibration period of the atoms, and A U the heat of
passage from one site to-an adjacent vacant one,
D.
2
U S tU
e- kT
/
-6t;'
Thus self-diffusion in crystals must give rise to a viscous flow in which
the velocity (strain rate) is proportional to the stress.
Such a flow is
characterized by the existence of a viscosity coefficient which is constant.
It
has the value
i-1
kT/D
Ordinarily this type of flow is masked by plastic flow, but it may be
observed when the stress is very low, as in the cases of surface tension
or residual stresses.
To make calculations based on this viscous flow, the energy
32
-
-
dissipated in flow per unit time is equated to the loss of surface energy
per unit time.
The latter is the product of the decrease in area by the
The former is twice the viscosity coefficient times the
surface tension.
volume of material flowing times the velocity of that material squared.
The problem is to find the velocity at any point of the volume and to sum
Mathematically, the summation of the
the energy all over that voluem.
tensor
Vik
=
j
xk*
(vi/
vk/ xi) is to be performed over the volume;
the x's are the three coordinates chosen, and the v's are the velocities
Then the energy of flow is
along those coordinates.
21
iVik
Frenkel shows how to perform this calculation for a single spherical pore
in a body of incompressible metal.
If the center of the pore is taken as
origin of coordinates, and spherical coordinates are used, then the only
The energy is
component of the tensor is Vrr = %v/Sr.
Cm2 2
2
Vrr Orr dr
The component Vrr
if a is the radius of the pore initially.
Sv/Sr
if found from the fact that the same amount of material must cross concentric spherical shells of different radii in the same length of time, dt.
At a radius R that amount is
=
4'WR2 dR/dt.
c= 4'\\ja2 da/dt so the velocity at any R is
surface of the pore,
V = dR/dt =
a2
R
Then
Vrr
dv/dr
At R a a, that is, at the
-
2a2 da
R3 dt
and the energy expended in flow is
da
dt
-
33
-
Equating this to the energy loss by reduction of the surface when the
da/dt: Since both T and
are constant, it follows that the radius of a pore
changes at a constant rate.
t =
ja
.
radius of the pore changes by da/dt gives
The pore then disappears after a time
Obviously this analysis holds only when ao?
lattice constant.
,
the
In Part III this analysis is applied to the case of
many pores in a metallic body, and also to various cases in which there
is gas in the pores.
The gas may be in equilibrium with a compound, and
there is then a constant pressure of the gas in the pore.
Or it may be
inert to the metal, so that it remains; as the pore decreases in size the
gas then is compressed until finally the pressure of the gas and the equivalent pressure of surface tension are equal.
Finally the gas may diffuse
out through the metal.
Frenkel also shows how this method of calculation may be applied
to the coalescence of two spheres.
Part III also contains a calculation
of the sintering rates based on this method.
In Part II, C, a critical examination is made of the whole question of flow, and as a result it is concluded that Frenkel's and Pines'
mechanisms are equivalent and are the only type of flow that can lead to
the disappearance of a spherical pore.
The other methods whereby material
can be moved from one place to another are evaluated and it is concluded
that in certain cases spheroidization of the pores is possible before completion of sintering.
In other cases sintering is complete before sphe-
roidization has proceeded appreciably.
34-
-
II.
Theory
A.
The Force of Adhesion
It has been brought out that modern investigators apparently
agree with Jones (Jol) that "the conditions which give rise to the sintering of a metal powder are identical with those which condition the cold
welding of massive materials."
It is appropriate, therefore, to review
briefly here the manifestations of this- cold welding of massive materials.
It may be mentioned that the cold welding is not restricted to metals but
is observed also in mica (Macaulay, Ma25), in glass optical flats (Rayleigh,
Ra6), glass beads (Stone, St24), quartz and sodium pyroborate (Bradley,
Br18) and between metal filings and glass, or porcelain (Beilby, Be24).
In all
these cases there has been demonstrated between surfaces a cohesive
force of magnitude ranging from a few dozen pounds per square inch to 5000
pounds per square inch, provided the surfaces are extremely clean and are
made to approach one another closely.
In the same way metallic surfaces have been made to adhere to
each other by simple approximation.
Tomlinson (To2) showed that freshly
cut surfaces of lead adhere when brought together again.
Freshly prepared
tin filings loosely piled in an evacuated glass tube are found after a few
days to be slightly adherent (Jol).
Most of the work done on the cold
welding of metal surfaces is reported by investigators of friction.
Perhaps
the most convincing demonstration of cold welding is found in a paper recently written at M. I. T. by Sakmann, Burwell and Irvine, who, making
their measurements by means of radioactive indicators (Sa8l, succeeded in
showing that surfaces brought into contact with no lateral motion whatever,
at least as far as the most careful experimental technique can assure,
35
-
-
adhere in places to such an extent that when the surfaces are again
separated there has been a transfer of material from each surface to the
other.
This establishes at once the fact that the magnitude of the force
of adhesion existing between surfaces brought into contact is of the order
of the cohesive force bonding the atoms together in metallic lattices.
It has been further demonstrated that the force of adhesion
perseveres to a considerable extent at some distance from the surface.
Schnurmann and Warlow-Davies (Sc33) have shown that it acts through lubricant films, causing sliding friction to occur by jerks.
They demonstrated
at the same time that the force is of electric origin by showing that two
metallic surfaces separated by a film of lubricant act like a condenser
which discharges between jerky motions and charges again during the movements, when the dielectric layer is thicker.
Jones finds evidence of the
distance of action of the force of adhesion in the indisputable coalescence of fine powder particles into groups some 400-700 Angstr'ms in diameter during sintering.
On this subject as on many other aspects of the sintering problem
there have been many hypotheses.
Spring (Sp7) claims to have liquefied
several low-melting metals by pressure alone.
Although it
is manifestly
impossible to do this by hydrostatic pressure with most metals (for which
the solid is denser than the liquid), yet if the liquid is free to escape
it is possible that this may be done.
Now since many manifestations of
cold welding involve pressing the surfaces together, it has been suggested
(Wr2) that any cold welding involves the formation of some liquid phase
between the two contacting points of the surfaces.
Jones (Jol) and
Rhines (Rhl) in their reviews agree that it is in no case necessary to
36
-
-
have recourse to a liquid phase in explaining cold welding; and the experiments of Sakmann, Burwell, and Irvine show that adhesion to the full
extent of the lattice bond is possible without appreciable pressure.
Others (Se9) ascribe a partial role in cold welding to the
vacuum formed between the surfaces when the air is squeezed out.
Holm
and Kirschstein (Hol6) demonstrated strong adhesion between nickel and
platinum when both bodies are in vacuo.
That vacuum plays a part in the
adhesion of optical glass flats, and of Johannson gauge blocks wrung together, goes without saying, but since vacuum is measured by the pressure
of the air on the exposed surfaces of the glass or steel pieces its contribution cannot exceed the atmospheric pressure, namely, 14.7 pounds per
square inch.
It might be thought that gravitational attraction could contribute a substantial term to the attraction of very small particles.
Hertz (He24) has shown that in the case of spheres of the size of the
earth the gravitational attraction is such as to cause a pressure to exist
at the contacting area in excess of the compressive stress at failure for
steel, i.e. many thousands of pounds per square inch.
He shows, however,
that the force per square inch on the area of pressure of two spheres in
contact due to gravitation decreases rapidly as the radius decreases.
Since
two balls the size of marbles do not adhere appreciably, it is concluded
that particles the size of metal powder particles will adhere even less
under the influence of gravitation only.
In view of this evidence many investigators have come to the
conclusion (Rhl) that the force involved in cold welding, or the cold
adhesion of powder particles or polished surfaces, is the same force which
37
-
-
binds atoms together into the lattice arrangement in which they are found
in metals.
Were there no interference from adsorbed gases, foreign films
of one sort or of another, and from irregularities in the surface on an
atomic scale (Rhl) as well as on a large scale it would presumably be
possible to cause metal surfaces to conform to such an extent that the
strength of the bond between them would rise to the value of the bond
across a grain boundary in a polycrystalline metal.
If the plasticity of
the metal is sufficient this is indeed possible, as in the compression of
filings of soft solder into a compact having very nearly the properties
of the cast metal (Shl2).
Wretblad and Wulff (Wrl) indicated the way in which a quantitative estimate of the extent of this force can be calculated.
They show
that as atoms are brought together from a great distance the force field
between them is at first an attraction.
Seitz (Se7) shows that this
attraction is chiefly of an electrostatic nature.
When the distance be-
tween atoms becomes of the order Qf a few Angstroms, another force, this
time of repulsion, begins to diminish the effect of the rapidly increasing
attraction.
This opposing force is due principally to the repulsion of
the ions left when the outer electrons of the atoms merge into the cloud
of free electrons which exists within the metal lattices.
Since the
force of repulsion is of extremely short range, it is possible to neglect
it in a calculation of the force of attraction between two approaching
surfaces of metal.
In the following pages the calculation is made of the
force of attraction between two copper surfaces separated by distances of
from 30 to 240 Angstroms (3 x 10'
to 2.4 x 106 cm).
Consider the Sommerfeld model of a metal, namely, an array of
38
-
-
point ions surrounded by a uniform field of free electrons.
For copper,
Huntington and Seitz (Hu5) have calculated that the field of free electrons has a density of one electron per atom, very nearly, and that these
electrons originate in the top two bands in the energy level diagram.
Two plates of infinite area and great thickness are brought together,
their plane surfaces being kept parallel, until the distance between them
is r.
The two plates are originally at the same potential (if
soon will become so (Fa4) ).
not, they
Some electrons are emitted from both these
copper surfaces by thermionic emission at any temperature above absolute
zero.
Each electron is attracted to both surfaces by a force known as
the image force, originating in the fact that the potential of the electron and that of the surface it has left behind are equal and opposite;
and so the electron is attracted by a force equal to that which would
exist if another electron of opposite charge were located symmetrically
to it with respect to the surface.
In turn, each image is attracted to
the opposite surface by an image force, and a system of such images is
immediately built up behind each surface.
Since both surfaces are attracted
simultaneously to the electron and to the images of the electron, it follows that there is a net force of attraction between the surfaces.
This
is the field of attraction of electrostatic origin described for one atom
by Wretblad and Wulff (Wrl) in a form in which it
can be calculated.
To
simplify the calculation, we note that Frenkel has shown (Frl6) that the
secondary images only contribute to the attraction a term about 1/10 of
the contribution of the electrons, and proportional to it.
It can there-
fore be neglected with the understanding that the result will be too small
by about 10%.
The force of attraction between the two surfaces may be
39
-
-
written as a negative pressure, and is the sum of the forces of attraction to the two surfaces of all the electrons between them.
Since the
force of attraction is proportional to the inverse square of the distance
between the electron and the image, the factor of proportionality being
the product of the two charges (negative for the electron, positive for
the image), we can write the potential energy of the electron at a distance x from one surface, (r-x) from the other, the two surfaces being
separated by a distance r; e being the charge on the electron,
V(x)
.
V1 (x)
2
2
V 2 (x)
()
+
4x+ --. (r-x) +e
in accordance with Seitz (Se7).
This function has the property that its
derivative with respect to x, that is, the force of attraction, is as
described above, and that at x
=
0 it reaches the value (-W -
4
e,
the work function for the electron for the particular metal corrected
for the influence of the approaching surface in lowering that work function.
This correction factor can easily (Frl6) be seen to agree with the facts,
for when r becomes infinite the factor is zero, that is, the work function
.
of the metal is (-Wl)
The number of electrons in a unit area of a layer of thickness
dx at a distance x from one surface is, at any temperature T,
dn = B dx
e
which is the Fermi distribution law.
I
+
(2)
B is a constant of normalization
which must cause the number of electrons at a distance x = 0 to be the
density observed inside the metal itself. E is the kinetic energy of the
- 40electron; k is Boltzmann's constant.
Figure 6 shows a schematic energy-level diagram of the two surfaces approaching one another when they are separated by a distance r.
The origin of the distance scale x is one of the surfaces, in this case
the one on the left.
The ordinate shows the potential energy of the elec-
The zero of potential energy is taken at infinity.
trons.
to it the potential energy of the electrons in the metal is
With respect
-
3, the
thermionic work function., Wa is the potential energy of the electrons of
zero kinetic enel-gy, and the difference Wa -
=
i is the average kinetic
energy (Fermi energy) of the electrons in the metal.
It is also the elec-
tronic work function for the metal because only electrons with energy
above this level are able to come out of the surface.
atomic distance is
The normal interV=O
_4
AVERAGE POTENTIAt
If
ACTUAL
i
METAL
FACES
a.
W
r
I
Figure 6.
Potential energy-level diagram in the vicinity of
two surfaces 1 and 2 separated by a distance r.
of symbols see accompanying text.
For explanation
41
-
-
We may now write the negative pressure (force of attraction
between the surfaces as
Pi
x r dV,(x)
dx
=0
dn
x r dn dV(
x=0 dx
It must not be forgotten that the opposite surface of each approaching
piece of metal, that is, the surface not involved in the attraction,
is itself subject to attraction to the electrons which are being emitted
from it.
P
This introduces the equivalent of a positive pressure equal to
for r = ow.
Furthermore, there is a kinetic pressure term, positive,
which enters in from the fact that the surfaces are continuously being
bombarded by the returning electrons and by those originating in the other
The value of this kinetic term is
surface crossing the whole distance.
found by treating the electron cloud as a gas in equilibrium with the
metal surface.
Slater (S15) shows that the pressure of such a gas is
T
Pk
)
e) kTi o N08NdT
dT j
T NINp
5
V--iT5/2 e,
Wa - Wi =where
the thermionic work function; No is Avogadro's number;
-
C
is the electronic specific heat; N, the number of electrons.
electron gas, if h is Planck's constant
i
2(Ztrm)
k5/2
h3
where m is the electronic mass.
2
Assuming no thermal expansion, C
where W
Cv
Nk
is the Fermi energy,
W,=
12m
(
8-"
2/3
Wi
For an
-
- 42
so that
T2 (W ) 2/3
=2N 1/k
;"1?k2 T 2m (8hV )2/3
.
k2T
6W.
T
dT=
26W
2 2 2
TdT = NOT k T
12 1
%
No
0
T
0
2T
2
T
dT
dT
0N
0
-
dT
o
12N kW~
12 Wi
Therefore,
k
T
kT12 Wj
)
M) 32k5/2T 5/2
P
e
In the ranges of temperature and work functions we are interested in, that
300 0 K T
is,
1 electron-volt> %
7 e.v.
2500 0 K
1 e.v. w )l0 e.v.
since k, Boltzmann's constant, is 0.863 x 10'4 electron-volts per degree,
=
"(min)
kT
o0.863 x 110-4 x 0 2500=
x 0.863 x 10-4 x 2500 - 0.17
(max)
fkT-12
5.00
12
i
so we can say that
12
Wi
The constant has the value
2(2%Wm)3/2 k5/2
h3
2(2-tW'x 9.1 x 10-28)3/2 x (1.38
(6.61)3 x 10-8 1
2 X0,33 x 039 x 2.22 x 10-40
2.89 x 10-9
x
10-16 5/2
= 0.665
So
43
-
-
Pk = 0.665 T5/2 e-q/
(5)
For values of T of 3000K (room temperature) and 20000 K (past the melting
point of copper) and for values of
of 1 and 10 e.v. (copper in various
stages of cleanliness and purity may have values in the vicinity of 3-7.
(Se7)), the following values of Pk are found:
T
300
1
10
Pk(dynes/cm 2 )
T
1.3 x 10"11
2000
1.3 x 10-163
Table 2.
2Pk(dynes/cm2
1
3.6 x 105
4
9.2 x 10-3
5
2.9 x 10-5
10
6.0 x 10-18
Kinetic Electron Pressure
At low temperatures and for values of g
over about 3 e.v., this pressure
is negligible, since only one of the above figures (T = 2000,
f
1) gives
a pressure near 1 atmosphere (106 dynes/cm2 ).
Vhen the distance r between the two surfaces becomes small,
that is, of the order of magnitude of the interatomic spacing, a term of
repulsion enters into the potential.
After Grz!neisen, it may be written
V'(x) . b/xn
V/S T)p
Cp( SV/ SP)T
where n :-6V (
2
V being the molar volume.
The constant b is found from the fact that the
total potential is a minimum at r
=
,
the interatomic distance; this
gives
b=
2
4e
ntl
(n(4 + e2 /Wi) 2
This potential is, however, neglected in the derivation of the Fermi distribution based on a uniform electron density.
So it must be neglected
-
- 44
here, keeping in mind that at
x when r falls to a few Angstr'ms the
attraction falls to a low value.
The next step in the estimation of the negative pressure of
Equation (3) is to evaluate B, the normalization constant, and the integral
of the Fermi distribution.
-V(x)/kT = i1..
In Equation (2), let C./kT
E, and let
Then
if l EB(kT)3/2 F
(
dn/dx = B (kT) 3/2
There is no analytical solution for F(V ), but it may be estimated, as
Stoner shows (St22), by means of series.
One series is adequate for the
region r (,0.36, the other for the region\(.2.3.
there is no satisfactory series.
Between 0.36 and 2.3
Stoner shows, however, that the deriva-
tive is continuous in this region, so that a smooth curve drawn graphically
in that range is a good solution.
For the region \k4.5 three terms of
the series shown below as Equation (7) are required; in the region
2.3<%\ -(4.5 only two terms are needed.
Fi(
) =(t_/2Ye- e2'/23/2
t
The two solutions are
e 3 y/33/2
-
... )
(6)
and
n
2
in which C 2 "2/12,
3/2
( +2
3~h(1
C4
=
0 2n
xx
2
.... ( -2n)
2
22n(7
)
(7)
0.947, C 6 = 0.985.
As has been mentioned, copper has very nearly one electron per
atom in the solid state, so that at x = 0, that is, right at the surface,
dn/dx = 1 electron per atom.
There are 4 atoms per unit cell of volume
46.97 x 10-24 cm3 , so
(dn/dx)o
=
4 x 1024 / 46.97 = 8.52 x 1022
-
- 45
This must equal
F.947
9,r3/2 1
3x2x3 .,. 4.987x3x3x5x7
B(kT)3/2
3
At x
.12X4
2x2x2x2\t
=
0, from Equation (1), \-
kT
2x2x2
= L(. e2
kT(4r,')
kT
Wx2
-b
2
4r+
1,277 x 102
B =
fit
ez
3/2
2
1.06+97
/4r+e
The values of B are calculated in Appendix III for the various values of
temperature T and of electronic work function Wi for which we are going
to estimate the attraction Pi(r)-Pi( tn).
These temperatures are 300 0 K
(room temperature) and 10000K (near 85000, the temperature at which the
compacts are sintered in Part IV).
tron volts.
The work functions are 1 and 5 elec-
The value of the force of adhesion is to be calculated for
distances separating the surfaces of 30, 60, 120, and 240 Angstr'ms.
Since the series of Equation (7) cannot be estimated at r =o-o , a distance
of separation of 10000 Angstroms is taken as representative of infinity.
As will be seen from the curves obtained, the error is negligible.
The values of r corresponding to the limiting values of
beyond
which Equation (7) is invalid, and those corresponding to the limit of
for which three terms are required are next to be found.
To do this, V(x)
is set at its maximum value, V(r/2), and r is then estimated from the
relation between V and\X
V(r/2)
=-kT
rmax
e,
'k
2
a -2e 2 /(2r + e 2 /i)
2
e
r
-
- 46
for y= 2.3(A)
. 4.5 (Angstroms)
rmax for
Wi(e.v.)
ToK
1
300
236
117
1
1000.
65
30
5
300
242
123
5
1000
71
36
Table 3.
Maximum Values of r in Equation (7)
We can therefore evaluate Pi for values of r ranging from 30 through 60,
120, 240 Angstr'ms for the various cases.
This gives us twelve points
-
on the curves of adhesi e f'o~rce ye
Table 4.
W
T
r
1
300
30
3
1
300
60
3
1
300
120
3
1
300
240
3 to 117; 2 to 240
1
1000
30
3
1
1000
60
5
300
5
300
60
3
5
300
120
3
5
300
240
5
1000
30
5
1000
60
(
No. of terms in F
3 to 30; 2 to 60
3
.30
3 to 123; 2 to 240
3
3 to 36; 2 to 60
Values of Wi, T, and r for which Equation (7) is valid.
For these ranges, then,we can write
Fjty
2 \ 3/2
2 x 3 9
123Q2.2 \
+ 2x.947xx
: 0.667\3/2 + 0.882V-1/2
2x2x2x2\t
,
22xx x2x
2Wx2xW
0710C-5/2 j 6.4641A-9/2
l7
6
47
-
-
-4 Be2(kT)3/2 = A, the equation for Pi reduces to
Writing
Pi = A
, _(
y = 4x + e 2 /W,.
To evaluate the integral, let
e
K
)
2
e2 I4r + 2
j -y t y
22e 2 )
C 1,
y
)
(4r+
Pi transforms to a five-
becomes
The parameter
term expression in y.
-(
) dx/ (4x + e2 /Aj)2
1:2
2e 2
2C
+r 2ee
Wiy (4r+Le)y ..
i
Cyy
Wi
if C is defined as
C
2e 2
4
Wi
The constant multipliers of the terms are written
D
D
D
2
0
.
1
-A
4+
\kT
Y
x 0.822 ( e2,,~
4
kT,
AAx0,710
e2 c 5/ 2
\kT
A x
I.6
3 -4
D
. 66 7 (e2C/2
9/2
The limits of integration are x a 0 and x = r, which, in terms of y, are
Y1 =
e2 1
2 = 4
e2/w1
For the four cases in which two terms must be used there is the additional
limit
Y2 : 4rm
+ e2/V11
Then Pi may be written
Pi = Di Y2 dy/(y2(Cy-y2)3/2
)71
-
- 48
D2
S2
dy/(y2(Cy-y2)-1/2)
yl
+ D3 Y2 dy/(y2(Cy-2)-5/2)
+D/
Y2 dy/(y2(Cy-y2)-9/2
)7y1
dy/(y2(Cy-y2)-9/2
-
4 cases only
Using the further transformation u = (C-y)/y, then c-y = uc/(u+l);
2
y - c/(u+l) and dy = -c du/(u+l) . The expression can now be integrated
by reduction formulae, and yields
2D 2(tan-lJ - uJ7)
+ D4 e8 U/
L
2 (21
2
u172)
93u.
+ 163u 2 - 21 g2)
-
+
4.
+
Pi = ( = 0.4 u5/2 - 2u3/2 - 6ul/2
(C4
896(u4l)8
15u3/2(l+ 29 u + 15 u2 + 3u3
+
164 (u - tan-lu
-3/
-
16384 (u+l)4
(5 + 17u + 75U2 + 15u3) + 150-
tan-lu
Y, or y2
Using the following values:
(S15)
e = 4.803 x 10-10 esu
k = 1.381 x 10- 16 erg/deg K
1 e.v. = 1.601 x 10'12 erg
Equation (8) is computed out in Appendix III for the values of T and of
Wi meriaoned above, and for distances of separation of 30, 60, 120, and
-
- 49
240 Angstr~ms, as well as for 10000 Angstrums.
The first set' of calcula-
tions in Appendix III show that the 3d. and 4th terms in Equation (8)
are smaller than 0.1% of the 'shole; that the upper limit values of terms
.
1 and 2 are less than 0.5% of the whole except for the term (2/ui).
Furthermore, the limitations imposed by Stoner's series analysis on the
distance in which his series is valid are removed, making it possible to
calculate the value of Pi for 10000 A without having recourse to graphical
or numerical integration.
The second set of calculations are based on the
simplified relation
5/2
i=--(0.4ui
c4
3/2
+ 2u1
+8
tan
-2D 2 (u-
-l
u
1/2
-8
1/2
U2)
4)
and include the calculation for 10000 Angstroms.
The last column gives
the values in pounds per square inch of the force of adhesion between two
surfaces Pi(r)
Temperature in
OK
oc
300
-
Pi(10000).
Work function
Wi in e.v.
Separation of
surfaces (A)
1
30
127000
60
64000
120
240
30
31000
16000
110000
60
30
60
52000
117000
48000
120
240
30
31000
18000
112000
60
44000
27
1000
727
1
300
27
5
1000
Table (5) below gives the results.
727
5
Table 5.
Force of Attraction
between surfaces, psi
The force of attraction between two
surfaces of work function W1, temperature T and
distance of separation r in Angstr~ms.
The same results are plotted in Figure (7).
-.
+I
4- T2
I~-
j-
-~t-
174;
4+U
441
or
--
c L--
Z:1I
Ifr
d-i-
Ky
-'-!{-
-50-
t;4-
-I
17 r~"Tj-
L
fa
4-
I7
7{
t-r'
--- jq
r.
HA--~
11,-
.,
4:
-i
4..
7:::J.T1
Li-,
.1
r
-7
i
-.--
51
-
-
It is evident that these results indicate forces of adhesion of
the order of magnitude of those observed.
The figure shows also that an
increase of temperature from 300 0 K to 1000 0 K causes a slight decrease in
adhesion, of the order of 10-20%, and that this decrease is less at higher
values of the work function.
This might at first seem to contradict the
results of Baukloh and Henke (Ba3l), who measured the adhesion of the two
halves of cut tensile specimens rubbed together under pressure and then
heated for 2 hours at various temperatures in various atmospheres.
They
show in all cases increase of adhesion with increase of temperature of
heat treatment.
Clearly, however, as in the case of the sintering of pow-
ders, the increase merely reflects the increased rate of flow of the metal
under the influence of the nearly constant force of adhesion.
They found
that the adhesion became measurable at a lower temperature for the speci-
mens heated in vacuum than for those heated in hydrogen or nitrogen in all
but two cases (the Cu-Ni and Cu-Fe combinations).
According to Chaffee
(Chl5) and Uhlig (Ulo) the number of electrons emitted from the surfaces
increases if they are covered with small quantities of a more electro-
negative material (oxygen or phosphorus on tungsten) and decreases if they
are covered with more electronegative material (caesium on tungsten).
It
is, however, probably unprofitable at this stage of the analysis of adhesive force to discuss in detail the effect of factors which can change its
magnitude by only a few percent.
Let us rather see what bearing the results found above can have
on the nature of the flow.
To do this, a calculation is presented below
of the stresses set up in two spheres closely approaching one another and
subject to the force calculated above.
An approximation of the data of
-
- 52
Table (5), which is good for all temperatures and all work function values
to about 10%, is
Table 6
Separation r
of surfaces in A
Force of Attraction
in p.s.i.
30
120000
60
60000
120
30000
240
15000
F
These values are fitted by a hyperbola
F r
C
If the separation is reduced to inches, C = 1.42 x 10-2.
In Figure (8)
below, two spheres are brought together until their nearest points are
separated by a distance S of the magnitude of the interatomic distance.
Due to the force F the spheres are drawn together until there is an area
of radius d over which the force of attraction F is balanced by the repulsive force due to ion repulsion (see above).
Then the force of attraction
f between the spheres is the sum of the force of attraction between elementary areas of cylindrical shape, thickness dy, and radius y, with y varying
from d to R, the radius of the particles.
f
S dR 2W'y dy
r
-
.1
x 10-2
The length of the cylinder, r, is found from the equations of
r
so that
= 2R - 2 R2
-
the circles representing the spheres to be
-
- 53
f
Let x =
2.836 x 10-2 -W R
2R -
V J -2
then f transforms ini o
0x
f
1418 xlC2WR
+,S
dx
(1+ E/2R) - x
-
R
-1.418 x 102'I R(
R
+ (1+
ln(l+
/2R)
/2R
-R-d
WhenS(4,R, as it is for particles larger than about 0.001r, and when
r
x
-
d
S
,
Figure 8. Two spheres separated by a minimum distance
subject to attraction across elements r of cylindrical
shape and thickness dy.
d is reasonably small with respect to R, this expression becomes
f = -1.42 x 10 2 R'W(l + 2.3 log(e
2R2
(9)
+ i))
2R
This force acts as a compressive stress across the column of radius d.
Introducing the area'\'d 2 , the stress is
1.42 x
-2 R- (1 4 2.3 log (
d2
2R2
+
2R
))
For particles of radius 0.004", such as are used in the experiments
(10)
-
- 54
described in Part IV of this thesis, and if S is 10 Angstroms (4 x 10-8
inch), S /2R - 0.5 x 10-5; for values of d/R of 10-1, 10-2 and 1c-3,
Equation (10) gives compressive stresses of 6.7 x 102, 2.3 x 10 , and
3.1 x 107 pounds per square inch.
At room temperature the yield point of
copper that has been s lightly work hardened is of the order of magnitude
of 104 pounds per square inch, so that the force of attraction causes the
spheres to form an area of contact of radius about 1/100th that of the
radius of the particle itself.
'X separate two spheres in contact over
that area requires a tensile strength of bout 40000 pounds per square
inch, or a stress in tension on the compact of a few pounds per square
inch.
It may easily be seen that the
the value of
\
has a very slight influence on
in this case as long as it is less than about 10 Angstroms.
The theory outlined above seems to be fully in accordance with
the experimental facts:
at room temperature, measurements of force of
adhesion should give results of the order of magnitude of the tensile
strengths of metals (as shown by the experiment of Sakmann, Burwell, and
Irvine (Sa8) showing transfer of metal particles from one member to another
upon contact).
The low cohesion of compacts of spherical particles is also
explained (compacts cake somewhat but can be broken up between the fingers).
The force of adhesion between particles is not sufficient to cause appre-
ciable contraction by plastic flow at low temperatures.
At high tempera-
tures the yield point of metals in general falls to less than a thousand
pounds per square inch, giving a value of d/R of about 1/10th, which
corresponds to a densification of the compact by plastic flow of
sin 0.1 = 0.1
or about 10%.
In other words, at high temperatures plastic flow might
55
-
-
account for a small part of the total densification.
A discussion of
this mechanism of sintering is given below in Part III.
Compacts under compression can be expected to show a much greater
adhesion because larger areas of contact are formed by the considerable
deformation occurring during pressing.
Although such areas are not neces-
sarily in the complete contact found for the spheres above, yet the distances of separation are probably of the order of a few hundred Angstr~ms
at most, and forces of adhesion of the order of 10000 pounds per square
inch are then predicted by the theory.
B.
The Surface Tension
It has been shown that, of the forces acting in bringing par-
ticles of metal together in a compact, those introduced by atmospheric
pressure (vacuum) and by gravitational attraction are negligible compared
with the electrostatic term in the cohesive force which exists between
atoms in the solid state.
The magnitude of this electrostatic force of
adhesion after the particles are in intimate contact has been left undescribed.
As two surfaces, which are not plane and parallel, are brought
together, there comes a time when parts of the two surfaces merge and
become one surface, or boundary; and at points where the transition occurs
between the region of two surfaces and that of one boundary the nature of
the stresses must now be investigated.
surfaces form an acute angle.
Clearly at such points the two
In this angle the emitted electrons set up
a force field tending to increase the area of the one-boundary region at
the expense of the two-surface region.
If the angle is thought of as
becoming more and more obtuse the force of attraction is accompanied by a
56
-
-
component which becomes increasingly parallel to the surfaces themselves,
as is evident in Figure (9).
B
a
Figure 9.
C.
A physical picture of the surface tension
term due to the electrostatic attraction.
In that figure, diagram (a) shows that when the angle is acute the force
is mainly one tending to close it.
When the angle, as in (b), is obtuse,
the force is resolved into two components, one pulling the metal outwards,
the other tending to close the angle.
Finally (c) shows that when the
angle is very obtuse the presence of the emitted electron sets up an outward attraction and also leaves a distinct component of tension in the
surface.
From this picture Gogate and Kothari (Go9) developed an expres-
sion for the surface tension of metals.
Their results show that the con-
cept is not sufficient to explain the surface tension.
Frenkel (Frl8)
and Dorfman (Do6) developed a simplification of the concept of an electronic double layer in the surface of the metal, including in their picture
other terms involved in the interacting forces present in a solid bounded
by an electron gas such as has been described.
Their analysis in turn was
superseded by an investigation into the nature of surface tension based
on the Gibbsian concept of a separate surface phase existing between the
solid and gas phases.
Section C.
57
-
-
This concept has been described above in Part I,
It supposes that in the surface phase there is a continuous
variation in density from that of the solid to that of the gas.
The sur-
face tension was shown to be related to the lateral potentials formally
written above.
Bakker (Ba35) described the origin of this surface tension:
if there were none, then the condition of mechanical equilibrium would require that the pressure perpendicular to the surface layer be equal to the
pressure tangent to the surface inside the surface phase.
These pressures
are not equal, and the surface tension is the summation or integral of
their difference along a path from the solid to the gas through the surface
phase.
That this difference must exist is shown by the fact that at abso-
lute zero, when the pressure in the gas phase is zero, that in the surface
phase must increase from zero to that in the solid.
At the absolute zero
there are no emitted electrons, but there is still surface tension, thus
showing that the electronic term which sufficed for the calculation of the
force of adhesion is not sufficient to deal with surface tension.
Accordingly Samoilovich (Sa9, Sa30) has calculated the surface
tension term at absolute zero from a different model.
He considers the
metal as an incompressible fluid of ions of constant charge density.
it
With
there is also an electron gas of constant charge density, terminating
at the surface with a density gradient dictated by the Fermi distribution
law, corrected according to modern concepts.
Setting up the system of
forces due to the kinetic energy of the electrons and the electrostatic
field described, he deduces the strain distribution in the surface, and
from the anisotropy of the strain tensor in the surface layer (where there
is electron density but no ions) he finds the expression for the pressure
-
- 58
tangential to the surface, and for the surface tension, in terms of the
electron gas distribution and the distribution of kinetic energy of the
electrons (Fermi energy).
It
is further shown that the surface tension
in non-metallic materials is arrived at in the same way, and differs from
that of metals in the presence of the strain-anisotropy term introduced
by the electron gas pressure.
It is this electron gas pressure term that
was found to be negligible in Part II above when dealing with the force
of adhesion between surfaces that are relatively far apart.
Therefore,
it is clear that there is no distinction between the force of adhesion
existing between surfaces relatively far apart and the surface tension of
the metal surfaces.
The two grade into one' another: in one the electro-
static term is more important; in the other the mechanical term assumes
predominance.
In between, when the surfaces are very close together, the
force field introduced by their presence is a combination of a force of
attraction and a tension in the surface layer:
the surface layer is, as
Gibbs showed, a continuous change from solid to gas, and has a very finite
width of several hundred Angstroms.
So two metal particles begin to
"touch" one another when their outermost ion layers are still far apart.
The numerical calculation of the surface tension in solid metals is cal-
culated in Samoilovich's paper.
tribution in the surface.
This involves knowing the electron dis-
He assumes that the kinetic energy distribution
is unaffected even inside the surface by the force terms (exchange and
Weizs"acker terms) due to the fact that the ions are not points; and there-
fore makes use of the uncorrected Fermi equation.
The density must satisfy
Poisson's equation (relating charge density and potential), so that he is
59
-
-
able to write the surface tension in terms of the potential inside and
outside the surface, much as has been done above in the simpler case of
the force of adhesion.
The result is, if 3 0 is the electron density
(uniform) inside the metal, in atomic units (1.31 x 10-2 for copper), corresponding to the number of free electrons per atom in the metal,
3/2
0
7/6
=030
Making the same calculation in terms of energy rather than potential, and
instead of the number of free electrons per atom (one for copper), solving
for the lowest energy for the metal, he obtains
4;S :: 0.285
7/6
0
3/2
- 0.64+30
The two solutions yield respectively 1128 and 1285 dynes/cm for the surface
tension of copper.
This is the tension at absolute zero.
Now the measured
surface tension of molten copper at the melting point is given as 1103
dynes/cm in the International Critical Tables.
From the fact that the
surface tension measured on liquids has a very small temperature coeffi-
cient, it seems reasonable to accept a small variation for the solid.
It
has been shown that the electrostatic term decreases slightly with increasing temperature.
The pressure term should follow this trend also.
In this
thesis, therefore, a value of the surface tension of copper of 1200 dynes/cm
has been selected, and it has been assumed to have no temperature coefficient.
The evidence above shows that this is probably correct within 10%.
So far in this analysis it has been assumed that there is no influence on the force of adhesion or on the surface tension due to crystallographic anisotropy.
This is not strictly correct.
The analysis given
above shows that the surface tension is due to two terms, the electrostatic term and the term due to mechanical pressure of the electron gas.
-
- 60
The density of the gas has been assumed constant within the metal, so that
the electrostatic term, which is the main one in the force of adhesion,
is probably independent of crystallographic orientation.
But the mechan-
ical term is probably dependent on the distribution of the ions near the
surface.
This distribution has also been neglected in Samoilovich's
analysis, but this is only a first approximation.
Lukirsky (Lu3) has
shown that ground spheres of NaC1 assume what is apparently an equilibrium
shape which deviates slightly from the sphere, and in which the (111) axes
protrude most, the (100) axes next, and the (110) least.
The difference
in length of diameters of this sphere parallel to these various axes
amounts to only five parts in ten thousand.
at 720-7600C created no additional change.
Very long periods of heating
Similarly the experiments of
Daniel (Dal6) on tungsten points heated to high temperatures in vacuum show
rounding below the melting point but very little, if any, deviation from
the sphere in shape.
There was, however, in these experiments, a definite
orientation of the field emission of electrons.
These effects are so
slight that it is justifiable in these calculations to assume no variation
in surface tension in various crystallographic directions.
The effect of contaminations of various kinds on the surface
tension is a question which has received very little attention in the
literature.
Since the electrostatic term is the least consequential, it
is to be expected that effects found to exist in electronic emission are
less important in surface tension than in the force of adhesion.
The
latter was found to vary only a few percent when the work function varies
from 1 to 5 electron-volts, a very considerable range. .The presence of a
heavy layer of compound, such as a layer of copper oxide on copper, will
-
- 61
probably be of no appreciable consequence as far as the shrinkage of a
pore under the influence of surface tension is concerned, unless that
layer has greater rigidity and opposes the flow by mechanical support.
For the question of contamination by the introduction of foreign atoms
in solid solution has been discussed by Gibbs (Gi3) in the case of liquids,
but the fact that in metals the surface tension is predominantly an effect
of the electron cloud leads one to believe that the presence of metallic
elements in solid solution does not have much effect on the surface ten-
sion as long as there is no phase change and the metals have nearly the
same number of free electrons per atom.
As Samoilovich shows, metals
with high density have higher surface tension.
The form in which the surface tension appears in the theory of
Frenkel, mentioned in Part I, Section D, is as an equivalent pressure.
According to Gibbs (G13) this pressure is
P
- P"1
2 r/R
in which P' is the equivalent pressure, P" is the pressure of the gas in
the pore, and C is the surface tension.
R is defined as the total radius
of curvature, and in terms of the principal radii of curvature it is written
1/R = i(1Rl t 1/R2)
Since the pressure of surface tension acting on a pore inside the metal is
opposite in direction to the pressure of the gas, the total curvature will
be considered positive when the metal is convex.
The principal radii of
curvature are defined by Franklin (Fr19) as follows:
in the vicinity of
the point in question on a curved surface the surface may be approximated
by a second degree surface, called the osculating paraboloid.
If the
coordinates are rotated suitably, the equation of the osculating paraboloid
62
-
-
may be made to consist only of second degree terms in x and in y alone,
-that is, it may be written
=
-(a
x2 + b y 2
)
z
Then a and b are the reciprocals of the principal radii of curvature R1
and R2 . Bartell and Osterkof (Ba34) show experimental evidence that
pores and particles down to radii of about 50 Angstroms show a constant
surface tension and an equivalent pressure given by the law of Gibbs.
The magnitude of this equivalent pressure may be calculated here.
Taking the surface tension as 1200 dynes per centimeter, the pressure for
various values of R is given in Table (7) below.
It shows that the stress
introduced by the surface tension is of the order of magnitude of stresses
Table (7). Pressure equivalent to a surface
tension of 1200 dynes/cm for various values
of the radius of curvature
R (cm)
10l
10-2
2R
(dynes/cm?)
2.4 x 101
2.4 x 105
10-3
10-4
2.4 x 106
2.4 x 107
10-5
10-6
2.4 x 108
2.4 x 109
2~/R (p.s.i.)
0.348
3.48
34.8
348.
3480.
34800.
commonly used for plastic flow only in pores or on particles of radius
less than about 10-5 cm. or about 0.l1.
Above 10-3 cm., or 10, the
range of stress is reached which Chalmers used in his studies of microcreep (Chl4).
Porosities of less than 10-4 have very little influence
on the density in compacts made of powders in the particle size range
around 325 mesh, but in micron-size powders such as are used in the carbide
industry and in the tungsten filament industry equivalent pressures due
-
- 63
to surface tension of hundreds or thousands of pounds per square inch may
become important.
In this section it has been shown that the surface tension of
solid metals is calculable by evaluating two terms, one of which is the
term responsible for the force of adhesion, and treated in the previous
section, and the other a term due to kinetic pressure of electrons; this
second term was found to play a minor role in the force of adhesion.
Consequently the two forces are really only one, and gradually merge:
the particles may then be said to be "in contact" when their nearest ions
are still several hundred Angstroms apart.
This single force field is
shown to be considerably more intense than other force fields which may
exist in a powder compact, and is therefore the only one which needs to
be considered in the study of the kinetics of the sintering process.
Its
magnitude is shown to approach the stresses which usually produce plastic
flow in metals at elevated temperatures, but does not reach the magnitude
of the stresses necessary to cause plastic flow at room temperature.
C.
The Flow
It
has been shown that plastic flow by slip is possible to a
limited extent when two spheres of small radii are brought into contact,
provided the temperature is high (giving a yield stress less than about
1000 pounds per square inch).
This flow takes place under the stress
introduced by the force of adhesion.
The possibility of the shrinkage of
a spherical pore under surface tension by a process of slip will now be
investigated.
From Nadai (Na3) we learn that there is a region of radius C
around a spherical pore of radius a in a massive piece of metal such that
64
-
-
inside the region plastic flow occurs, and outside the region elastic
deformation only takes place, if C is given by
P = s. (1 + 2 ln C/a)
whence, using the Gibbs relation P
(11)
2wT/a
C ae aso
For a pore of radius
10
X cm, at room temperature, taking the yield stress
of copper for a slow rate of strain to be about 10000 pounds per square
inch
- 0.5)
C = 10~X e(3.01 x l0x-6
At the distance C, then, there is a deformation inward towards the pore
to the extent allowed by elastic flow at the yield stress, so that
dC/C
If
=
d Z_.= 1,49 x 10-12 x 10000 x
2.54 x 2.54
454
x 981
1.03 x 10-3
we consider the plastically flowed material to be incompressible
(this is done in Nadai's derivation of the equation for P above), then
the volume of material which has moved into the pore is
4'jC 2 dC = 4-\C3 dc/C = 1.32 x 10-2 C 3
(12)
Table (8) gives the values of C for various radii of pores.
It also
gives the values of 1.32 x 10-2 C3 , and compares these values with the
volume of the pores.
For pores of the order of 10'6 cm (0.017) or larger
the table shows that at room temperature less than one one-thousandth of
the pores can be filled by a slip process of plastic deformation.
Pores
smaller than this limiting radius are, however, filled, and can evidently
not continue to exist in metal at room temperature.
At elevated tempera-
tures the yield stress falls to values which depend considerably on the
65
-
-
rate of application of stress; therefore, it is difficult to tell what
part of a strain is due to slip and what to some other process.
If the
yield stress falls to 1000 pounds per square inch, the limiting radius for
slip becomes a little over 10- 5 cm.
At a yield stress of 100 pounds per
square inch the limiting radius is about one micron.
Table (8).
Radius of pore
10-1
10-2
10-3
10-4
10-5
5x10-6
10-6
10-7
Shrinkage of Pores by Slip (Yield at 10000 psi)
C
Volume of pore
4.2
4.2
4.2
4.2
4.2
5.2
4.2
4.2
10-3
10-6
10-9
10-12
10-15
10-16
x 10-18
x 10-21
x
x
x
x
x
x
1.32 x 10-203
6 x 10-2
6 x 10-3
6 x 10-4
6 x l0-5
8 x 10-6
5.6x10-6
1.3 x 10-5
7.4 x 10+5
2.9
2.9
2.9
2.9
6.7
2.3
2.9
5.4
x
x
x
x
x
x
x
x
10-6
10-9
10-12
10-15
10-18
10-18
10-17
l0415
The kinetics of such a plastic flow are readily studied: the
pressure equivalent to the surface tension increases as the pore shrinks,
that is, as the flow proceeds.
during the process.
Therefore, the flow rate must increase
This means that for those pores which only shrink
a slight amount the process is as rapid as is the measurement of a yield
stress:
it is over quasi - instantaneously.
For those pores which dis-
appear, the volume of displaced material is even less, since the pores are
smaller, and the process is therefore also instantaneous.
From a point of view of sintering times, then, it is concluded
that processes which take place by slip are of an instantaneous nature.
It must be emphasized, however, that the shrinkage of pores by this process
does not constitute sintering.
Indeed, it may be shown that the dis-
appearance of pores by this method does not change the density of a compact:
66
-
-
outside the radius C of the region of plastic flow the metal is only
elastically deformed, and upon cooling the elastic strain is merely increased to the appropriate value at the lower temperature.
Therefore, some
other process must be investigated to explain the disappearance of pores
during sintering.
This other process must furnish means of relieving the
elastic strain set up by the surface tension stress.
Plastic deformation
by slip is also unable to explain even the disappearance of larger sizes
of pores.
On the other hand, if sintering takes place by the coalescence
of spheres and is completed before the pores formed by the contact of these
spheres can become closed up, then the plastic flow process by slip does
constitute sintering and does cause the whole compact to shrink.
Since
it is an instantaneous process, it must take place during heating up, and
a compact heated up and immediately cooled again should show appreciable
shrinkage.
Some dilatometric curves made by P. Duwez (private communica-
tion) at California Institute of Technology on copper compacts alternately
heated and cooled do indeed show that some shrinkage has taken place during
the heating and. cooling cycle.
Since the work was done on compressed com-
pacts, his results do not lend themselves to calculation, and it is not
possible to say whether the shrinkage is due to the compacts having remained
an appreciable length of time at elevated temperatures, or whether it is
due to instantaneous plastic flow.
The influence of the transient effects
discussed in Part I, Section B, makes this question an extremely difficult
one to resolve experimentally.
In this thesis the experimentation has
leaned rather towards the question of showing whether or not the sintering
rates are consistent with the theory of viscous flow of Frenkel.
If the
-
- 67
rates found are more rapid than those predicted for viscous flow, then
it is probable that some instantaneous shrinkage has taken place during
the heating up period.
expand
As will be seen in Part IV, the compacts actually
appreciably during the first few minutes, and only subsequently
begin to shrink.
This swelling, which is due to the expulsion of volatile
matter, completely masks any flow by slip that may have taken place.
At elevated temperatures plastic flow can take place at much
smaller stresses.
The yield stress is not zero, however, for, if it were,
all pores, no matter what their size, would shrink rapidly under the influence of surface tension.
is a barrier:
In other words, for a process of slip there
If
if the stress exceeds this barrier, flow takes place.
it does not, flow never takes place, and any flow must be ascribed to some
other phenomenon.
In actual practice, of course, the temperature soon
reaches the recovery level, and therefore the presence of a yield stress
or barrier is hidden by that process.
Without going into the question of
whether the recovery is by diffusion or by some other movement, the other
modes of flow may be briefly investigated.
Secondary creep may occur at elevated temperatures.
This form
of flow also has a barrier, such that a very finite minimum stress is required before it can take place.
Kauzmann (Ka5) has shown that in any shear
reaction (any deformation in which units of flow, be they slip bands, blocks
separated by dislocations, or single atoms, go by one another) the rate
of the reaction is given by the expression
s,
CT e-F*/RT sinh (A%-/kT)
where AF* is the free energy of activation of the unit of flow,
(13)
I is the
stress, C and A are constants, and k, R, and T have their usual meanings.
-
- 68
If A
~ is much larger than kT, then
eA
sinh(A a-/kT)
/T
and s may then be written for small deformations
log s = B + B' T
where B and B' are functions of temperature,
(14)'
This is clearly the law of
secondary creep, and B/Bt represents the barrier stress under which no
flow takes place.
If, on the other hand, A W is much smaller than kT, then
sinh(A T /kT) '
A ~/kT
s = B" I
and
(15)
In this case flow may occur at any stress except zero, and there is no
barrier.
Flow of this nature has been observed experimentally by Chalmers
working with single crystals of tin (Chl4), even at room temperature.
Using optical interference methods of measurement, and carefully correcting his results for thermal effects due both to his apparatus and to the
stress itself, he was able to measure isothermal creep rates that were
constant and proportional to the stress up to stresses of 170 pounds per
square inch.
By the theories of Kanter (Ka8) and of Frenkel (Fr8) this
flow should be due to a viscous movement of single atoms at the rate which
is characteristic of those atoms in self-diffusion.
Kauzmann (Ka5) is of
the opinion that this flow is due to slip of blocks of atoms bounded by
dislocation planes, and takes place at the rate at which dislocations are
formed due to self-diffusion of the atoms.
If the former are right, the
rate of flow is calculable from the self-diffusion coefficient, and gives
a coefficient of viscosity
d! e
(16)
-
- 69
If Kauzmann is right, then the flow still has a constant coefficient of
viscosity, but it is
Q/kT
C (T) e
(17)
In these equations k is Boltzmannts constant, T is the absolute tempera-
ture, Do is the constant in the self-diffusion coefficient, Q is the heat
of activation of self-diffusion, S is the lattice parameter, and C(T) is
a function of temperature but not of time.
Kauzmann shows that creep in
lead is faster than is given by the self-diffusion viscous flow, as calculated by Kanter (Ka). There is no way of calculating his function
which is written here C(T), nor any way of telling whether it
is faster
or slower than the Frenkel type of flow, unless some way is found of calculating the rate at which dislocations are formed.
From this brief survey of the possible mechanisms of flow it
is concluded that any form of flow (microcreep, primary creep, secondary
creep, or slip) is possible in the sintering process.
Two of these forms,
slip and secondary creep, involve a barrier or lower limit of stress which
makes flow under extremely small loads impossible.
The other two, primary
creep and microcreep or viscous flow, are possible down to zero stresses,
but primary creep seems to be (Se8) a transient phenomenon, of limited
extent, which, although it might explain a part of the disappearance of
pores in sintering, cannot deal with the entire phenomenon for large pores.
The two forms of flow involving barriers cannot be used in studying the
disappearance of pores because, as was stated above for the flow by slip,
there is always a region beyond which the stress is less than the barrier,
and outside this region the metal is strained elastically only, so that
such a mechanism of flow, while it
can explain the filling in of a pore,
70
-
-
cannot thereby fulfill the condition of shrinkage of the compact.
There are left, then, only the two forms of viscous flow, that
in which the units of flow are individual atoms, and that in which the
units are blocks of atoms bounded by dislocations.
are formed by self-diffusion, it
If the dislocations
is expected that the rates of these two
forms of viscous flow are essentially the same.
Since Frenkel shows on
theoretical grounds the way to calculate the rate of this flow, it is
reasonable that the sintering process be studied along his model here.
If the results of the calculation show rates that are too slow, then it
is to be concluded that a dislocation mechanism must be resorted to.
In
that case no calculations of rates can be made.
It
has been shown above that plastic flow by slip may take place
in the course of shrinkage of metal powder compacts only if the densification of the compacts is more rapid than the spheroidization of the pores
between the particles.
It is therefore necessary now to calculate the
rate of this spheroidization.
Two mechanisms are possible:
diffusion, and evaporation and condensation.
surface
The former involves the jump-
ing of an atom out of the surface, its migration along the surface, and
its jumping back into the metal.
The second process involves a process
that is similar except that the atoms migrate through space instead of along
the surface.
Both processes require that the atom have free energy in ex-
cess of a minimum amount known as the activation energy.
It is generally
conceded that the activation energy for surface diffusion is less than the
activation energy for evaporation.
In both cases the energy of condensation
is equal to that of evaporation, so that no net energy is used up in overcoming the activation barrier; the minimum energy required is therefore
-
- 71
the infinitesimal energy to activate one atom.
The two processes can
therefore be studied together, and the two activation energies can be used
in turn in the resulting equations to give the two rates of rounding.
The
activation energy for the evaporation and condensation process is the
sublimation energy, known for copper to be 81500 cal./mol for a plane
surface; the energy for surface diffusion is not known for copper.
As
was shown in Part I, Section C, the free energy of sublimation increases
for small pores, because the surface tension contributes a pressure term
to the equation for mechanical equilibrium.
The maximum rate of transfer of material by evaporation and con-
densation may be calculated by means of the kinetic theory of gases.
assumptions involved are three in number (Kn6).
sidered ideal.
The
First, the gas is con-
In our case the gas is the vapor of the metal being sin-
tered, and at sintering temperatures the pressures of the vapor are so low
that the assumption of ideality is amply justified:
it is shown below
(Table (9)) that the pressure of copper in the pores at 8500C is of the
order of 10~9 dynes per square centimeter.
Second, the theory assumes that
the currents of gas are small with respect to the average velocity of the
atoms.
In the example below the current is of the order of 10-20 cm/sec,
whereas the velocity of the atoms of copper is, on the average,
6
14550
TAA = 14550 x 4.2
=
6.1 x l04 cm/sec
(18)
In this expression, T is the absolute temperature, M is the atomic weight
of copper .
The third assumption is that thermal equilibrium exists in the
region in which the gas is moving.
Since the calculations are based on
short heating-up periods and long sintering times, this assumption is
reasonable.
The theory shows that the number of atoms colliding with a
-
- 72
,
surface is-j N U , where N is the number of atoms per cubic centimeter.
2
The pressure exerted on that surface by the collisions is then 1/3 NmE
where m is the mass of the atom.
The average velocity is given by Maxwell's
distribution as in the equation (18) above; and from these expressions the
mass of material impinging on a unit surface in unit time is
G = 43.75 x 10-6
where p is the pressure.
/
p
(19)
The theory has been shown to be in complete
agreement with experimental fact as far as monatomic gases (like copper
vapor) are concerned.
Now if the pressure is small, as it is in our case,
then the flow of gas through a space takes place by "effusion" (Kn6), and
the amount of gas flowing past a unit surface perpendicular to the flow
lines can also be found from the considerations given above.
It
is, per
unit cross-sectional area across the lines of flow
G = 43.75 x 10-6 4T
(Pl-p2)
where the p's are the two pressures on either side of the surface.
If the
gas impinging on the surface of lower equilibrium pressure is completely
condensed as fast as it hits, then the rate of evaporation, once the flow
is under way, must be equal to the rate of condensation given above by
Equation (19).
Clearly the maximum rate of evaporation exists when the
pressure in the gas immediately next to the surface (pressure p1 ) is as
different as possible from the equilibrium pressure of gas above that
surface (pressure pel).
Similarly the greatest rate of condensation occurs
when the pressure p2 of the gas near the condensing surface is as great
as possible compared with the equilibrium pressure Pe2.
are equal, are maximum when p1 = p2 = p.
These rates, which
Considering the surfaces of
evaporation and condensation as apertures through which the gas is passing,
-
- 73
the rates of evaporation and of condensation are then
G
= 43.75
x 10-6qF7I
(PelP)
43.75 x 10-6 J
(20)
(P-Pe2)
This can be solved only when
p
Pel + Pe2)
and the rate of transfer of material from a surface above which the
equilibrium pressure is Pel to one above which it is Pe2 is then, per
unit area,
G
=
21.875 x 10-6
/
(PelPe2)
To calculate the two equilibrium pressures at two points inside
a pore where the radii of curvature are R1 and R2 , use is made of the
Kelvin equation (G13)
ln (p/p2 )=a M)
TRT
--
1l
(21)
R2
in which CI is the surface tension of the solid surface;
is the solid
If one surface is plane, R2 =0*m
density; R is the universal gas constant.
so that the expression becomes
ln p,
l n p am +
q
But from thermodynamics it is known that
ln p
so that
=
-AF 0 /RT
(22)
AF. 2aM
p1
ae T
(23)
It must be remembered that, in a pore, R1 is negative.
The rate of trans-
fer from a point of total radius of curvature R 1 to one of radius R2 is
then
74-
-
26M
2VM
AF
G = 21.875 x 10-6
e HT (erNi
(24)
-eF'2)
For copper, using the surface tension value TS
1200 dynes per
centimeter, a density of 8.99 grams per cubic centimeter, and a molecular
weight of 63.57 grams, the value of
C = 2 M/R = 2.039 x 104
0 cm
The specific heat (Cp) of the monatomic copper gas is 4.97
calories per mol.
That of the solid copper at temperature T is, from
Kelley (Kell), 5.44 - 1.462 x 10- 3T.
C
The difference is
= -0.47 - 1.462 x 10' 3 T
Using Kelley's values of heat of sublimation (A H29 8
and entropy of sublimation (I\ S2 98
H 0 = 81525 + 140.06
81525 cal/mol)
-31.83 e.u.) we obtain
+
65 = 81730 cal/mol
The standard free energy of sublimation is
= 81525 - 31.83 x 298 = 72040 cal/mol
F
298The constant of integration I is
I = 72040/298 - 81730/298 - 0.47 ln 298 - 0.731 x 10-3 x 298
= 241.745
- 274.202 - 2.678 - 0.218 = - 35.413
81730/1.987T + 0,47 ln T/l.987 t 0.731 x 10- 3T/1.987
AF /RT
-
35.413/1.987
-
Then the free energy of sublimation at any temperature T is AFT and
=
41132/T + 0.23654 in T t 0.36789 x 10' 3T - 17.822
(25)
Table (9) shows the values of this expression, and the pressures corresponding to it for plane surfaces.
Table (9).
75
-
-
The Equilibrium Pressure of
ToC
LFO/RT
p (dynes/-c
27
120
8 x 10-
400
45
53
3 x 10-20
5 x 10- 1 2
1.10 X 10-9
1 x 10-7
26
20.819
16
700
850
1000
)
Vapor over Copper at Various Temperatures
Consider a pore having two different radii of curvature.
For
simplicity, let the pore be a cylinder of radius Ro and length, say, 10RO,
capped by two hemispheres of the same radius as the cylinder.
It is easily
shown that the total radius of curvature of the cylinder is 2R0 ; for the
spherical caps it is R0 .
If the radius of a spherical pore of equal
volume is 10'2 centimeter, as is approximately the case in the experiments
described below, then
R
=
2
/
x 10
0.344 x 10-2
The flow is most rapid when the radii of curvature are most different,
that is, in the beginning.
G
21.875 x 1- 6
At this time the flow is
e-20 .819 (e.86x10/0.344xl0
63.57/1123
1.816 x 10~ 7/0.688x10-2)
(21.875 x 2.379 x 2.64
/
1.10) x 10-21
1.25 x 10-19 gr/cm2 see.
Since the surface of evaporation is much greater in this case than the
surface of condensation, the latter is the limiting one, and upon it
material is condensing at such a rate that the surface is moving towards
the center of the pore at a rate
dr/dt = -1.25 x 1019
/
8.99 s -1.39 x 10-20 cm/see
-
- 76
Now the Frenkel type of flow gives a rate, as will be seen in Part III,
for the spherical pore of equal volume, of
dr/dt = -3600/16 x 1.25 x 109
=
-1.8 x 10~7 cm/sec
From the development given above it is clear that if the radii of curvature differ even more widely the transfer through the gas phase is more
rapid.
But even a difference between a plane and a surface of radius 107
Surface diffusion is
can only bring the rate up to about -10'-14 cm/sec.
much more rapid than evaporation and condensation, but it is unlikely that
it is some 107 times as rapid, so that it can be concluded from the calculation shown above that in the case of copper at 8500C the pores do
not become rounded during sintering, at least not by either surface diffusion nor by evaporation and condensation.
that this is so.
The experiments of Part IV show
Spheroidization does not occur unless the pores reach
a constant volume by virtue of the entrapped gas.
This is probably the
in
case, and small spherical pores may be seen, specimens #3 and #12.
The
first was sintered in vacuum for 105 hours and contained entrapped hydrogen
at an initial pressure of 150 mm.
atmospheric pressure for 16 hours.
The second was sintered in argon at
Delisle (De2l) shows the formation of
spherical pores in compacts sintered from -200 mesh copper shot at 105000
for 3 hours in dry hydrogen.
The shape of the pore may or may not have an influence on the
rate of shrinkage by viscous flow under the influence of surface tension.
It is believed that the influence of shape is small.
vanced below for this belief.
Two reasons are ad-
In the first place the rate of shrinkage
of a compact may be looked at either from the standpoint of a large number
of particles coalescing into a dense mass, or else from the standpoint
-
- 77
of a mass of metal containing numerous pores which are disappearing.
This is illustrated in Figure (8).
The diagram on the left, (a), shows
a pore between six spherical particles.
That on the right shows the
same structure visualized as a pore in a mass of metal.
If we study the
OI
Figure ua. TEwd~Tiewpoints in the 5tudy or the
Shrinkage of Compacts: (a) Spheres Coalescing;,
(b) A Pore Shrinking.
rate of shrinkage of the two systems, using a spherical substitute for
the pore in diagram (b), and the two calculations give approximately the
same answer, it is then evident that a spherical pore and a pore of complicated shape shrink at the same rate.
This is shown to be so below.
The second reason for believing that the shape of the pore makes
little difference in the rate of shrinkage of the compact is this:
at a
distance from the pore great compared with its radius., the material is not
subject to the influence of variations in shape of the pore:- it only
responds to the average stress introduced by the pore in the metal.
Mathematically, the flow in a metal containing no sources or sinks is a
harmonic functionfor the metal is incompressible.
The presence of a sink
such as a pore can be treated as a local region in which the divergence
of the flow vector is negative.
But by the divergence theorem, as is shown
in any elementary- text in vector analysis (Ph2), the divergence can be
-
- 78
calculated equally well by means of the flow at each point along the
surface bounding the pore or else by means of the total amount of material
flowing into the pore.
So it is necessary to calculate the average stress
introduced by an odd-shaped pore at a large distance from the pore itself,
then calculate the dimensions of a spherical pore which introduces the
same average stress in the metal.
It is shown below that the volume of
the odd-shaped pore and that of the sphere turn out to be equal, so that
the influence of pore shape on rate of shrinkage of the compact is very
small, provided the pore does not depart too far from an isometric shape.
It was shown in the Introduction, in reviewing Frenkel's work,
that the rate of shrinkage of a pore such as that shown in diagram (b),
Figure (8j, is given by
dr/dt =
where r is the surface tension and-
(26)
is the viscosity coefficient (the
reciprocal of Kanter's"flowability" (Ka8)).
Using the same technique of
calculating the energy dissipated in flow and equating it to the lost surface energy to a system composed of two spheres in contact, we find that,
for each sphere, the loss of surface area is
so - S = i S
= 4\Wa2
=
2Wr 2 sin' d
- (2Wr2 +
4\Ya2 - 2\r2 (l
t
cos 9)
The symbols are those shown in Figure (5).
)
=
(27)
The starting radius is a;
and the radius after flow has begun, when the spheres are in contact over
a circular area as shown in the figure, is r.
But since r changes very
much more slowly than 9 at the beginning of the coalescence we can set
a = r, and the loss of surface is then
S = 2-Va2 (1 - cos 9)
x
a
x
IX
0-
-
C14
Figure (5).
The Coalescence of Two Spheres
The work of surface tension is
-6~dS/dt = 2-C a2 sin 9 dG/dt
For small angles 9, sin9'lt 9 , so
-(zdS/dt
2'tC a2 9 dQ/dt
(28)
The amount of flow energy expended is more difficult to calculate.
Frenkel attacks the problem as follows:
at the beginning of the deforma-
tion the distortion of the sphere is small compared to the motion of the
whole sphere in approaching its neighbor, so that the total flow is approximately obtained by having every point in the sphere A move in the x-direction
in Figure (9) by an amount
r - r cos 9 = 2r sin2 9 =-j rg
for small angles 9.
2
The rate of flow is then
dd( r 9 2 ) = r 9 d /dt
Then the tensor of flow Vik reduces to the component
(29)
-
- 80
V
=
-J
= 9 dG/dt
(2dv,/cbx)
The energy dissipated in flow is then, if V is the volume of the sphere
2r
dV =
\Yr 3
92 (d9/dt)2
(30)
Equating the
since the velocity is constant throughout the volume.
energy of flow with the work done by surface tension, and as before setting a = r for small angles of 9,
r3
2 (d9/dt)
- 2
2O(
2
a.
9 dG/dt
a 9 dG/dt =
Integrating,
92/2
-
4
(31).
a
At the beginning of the coalescence, then, the center of the sphere
approaches its original point of contact with sphere B at a rate such that
x = a cos 9 = a - a(l-cos 9) = a - 2a sin2 9
= a - a9 2 /2 = a (1 - 3Tt)
(32)
4ar\
During the time considered in the analysis above, when r changes much
more slowly than 9, it then follows by differentiation that
dx/dt = -34--/4
just as in the previous case (Equation (26)).
(33)
It follows that for the
complicated-shaped pore existing between six spheres there can be substituted a single spherical pore of radius equal in this case to the
radius of the spherical particles.
For other structures or packings of
spheres this may not, of course, be strictly true, and the radius of the
pore might not necessarily be that of the particle.
To show that it is,
rather than calculate the volume of the odd-shaped pore and the radius of
an equivoluminous sphere, it will be noted here only that in the experiments performed in Part IV of this thesis the starting apparent density
-
- 81
of the packed powder is very nearly 50% of the theoretical density of
copper.
Counting one pore per particle -- this is justifiable if the
size range of the particles is very narrow and the particles are wellpacked spheres -- the radius of the sphere equivoluminous with the average
irregular pore is exactly that of the average particle.
The second calculation showing the very slight effect of pore
shape on rate of shrinkage of the compact is the following:
a cylindrical
pore with a spherical cap at each end is investigated from the point of
view of the force exerted at a distance from the pore by each element of
surface, in the direction of the radius vector of that. element of surface.
This force is then summed over the whole area of the pore, and the result
is compared with the integral of the force calculated in a similar manner
for a sphere of volume equal to that of the cylindrical pore.
Let the
coordinates qcylindrical) be set up as shown in Figure (10).
If'b
Figure 10. The force exerted by an element of the surface
of a cylindrical pore in the direction of the radius vector
and at a distance R from its center.
82
-
-
Then the total force in the direction of the radius vector is
/2
2
tan~1 2
(radial stress per unit ofl)t
in cylinder)
- 2
2A
t)
(radial stress per unit b;Ck
in spherical cap)
0
This is shown in Appendix IV to be equal to
3
F =8
1 (1 + 2A)
(34)
4
R2
In the case of the sphere the total 'force is
'/2
2
F
2 r,
0
r.2'r2 sino.d co
8 lf
r
/R
(35)
In Appendix IV it is shown further that when the sphere and the cylindrical
pore have the same volume, r3 c r3 (1 t 4A), so that the two expressions
are identical, and for pore shapes that are not too far from isometrical
the odd-shaped pore may be replaced by an equivoluminous sphere.
It has been shown in this section that the sintering process
which takes place under the influence of the force of surface tension, or
of adhesion, shown to exist in the previous section, cannot take place by
a slip type of plastic flow.
Qualifications to this statement are:
(1) small spheres when first brought into contact deform to a maximum
extent of about 10% by plastic flow at high temperatures, and (2) at elevated temperatures when the yield stress falls to less than 1000 pounds
per square inch pores under 10-5 cm in radius disappear by plastic flow
and leave a region of plastic deformation surrounded by a region of elastic
deformation.
Plastic flow of this nature cannot, however, constitute
shrinkage of the compact, as its extent is restricted to small regions;
(3) some shrinkage by plastic slip occurs during heating-up, but is usually
masked by other transient transformations which also take place during
heating-up.
83
-
-
Secondary creep, like slip, is characterized by the presence
of a barrier, or minimum stress under which no flow takes place.
Such a
barrier causes flow to occur only in restricted regions around the smallest
pores, and some other mechanism must be found that is capable of relieving
the elastic stresses set up by such flow, as well as to explain the flow
into larger pores.
Such another process, which may be allied to recovery
in strained massive metals, is shown to be possible if the stress is much
lower than the activation energy of the units of flow.
In that case the
rate of flow is proportional to the stress, that is, the flow possesses
a constant viscosity coefficient, and may occur under any stress down to
zero.
Such flow has been observed by Chalmers in tin, and is shown theo-
retically by Frenkel and by Kanter to be a consequence of the existence of
self-diffusion in metals.
A similar viscous flow can probably exist in
which the units of flow are small blocks of atoms bounded by dislocation
planes, but until more is known about dislocation generation, no calcula-
tions can be made on that basis.
The self-diffusion type of viscous flow
lends itself to calculation -according to the method shown by Frenkel.
The question of the spheroidization of pores has been tudied,
and it is shown that under the sole influence of surface tension without
interference from gas pressures inside the pores the process of sintering
by viscous flow is much more rapid than spheroidization by evaporation and
condensation, and probably faster than spheroidization by surface diffusion.
Experiments described in Part IV confirm this view.
Finally the question is studied whether it is possible for purposes of simplification in calculations to substitute for irregularlyshaped pores spherical pores of equal volume.
It is shown in two ways that
this is indeed possible.
84
-
-
Calculations made in the next part are therefore
based entirely on spherical pores; the flow is assumed to have the property
of a constant coefficient of viscosity; and the force involved is assumed
to be only the surface tension, modified by the pressure of such gases as
might be present inside the pores.
Under these conditions flow takes place one atom at a time, and
its rate is closely lihked with the rate of self-diffusion.
Such a flow
has been observed to take place under a variety of very small stresses.
A concentration gradient of lattice imperfections introduces a free-energy
gradient; an electric field has been observed to induce electrolytic migration of carbon atoms in gamma iron and of gold in lead and palladium (Wal4);
and consequently a mechanical field of force should give rise to a flow
of the same nature.
Since in the other fields of force the rate of flow
is always proportional to the coefficient of self-diffusion and inversely
proportional to the temperature, as Einstein has shown (Se7) for the electrolytic migration and as is well known in the case of concentration gra-
dients, it is logical to accept Frenkelts view that the rate of flow under
mechanical stress is also proportional to these factors.
It should there-
fore be observable in sintering and in aftereffects after the introduction
of elastic stresses.
That such flow has a constant coefficient of vis-
cosity is shown by Kauzmann (Ka5) to be due to the fact that the activation
energy for self-diffusion is much greater than the stresses imposed.
85
-
-
III.
Calculations
A.
The Case of Vacuum Sintering
The first calculations are made of the rate of disappearance of
pores in the vacuum heating of a metal which is incompressible in itself,
but which, by virtue of the fact that it is full of pores, can be treated
as a compressible material at a distance from the pore.
face completely enclosing a region containing pores.
region consider a spherical pore of radius a.
Consider a sur-
At the center of the
The rate of flow of material
into the region through the surface is equal to the rate of flow into the
The net amount of metal crossing the boundary
pores inside the region.
of any spherical surface of radius R, concentric with the pore of radius
a, is then
4-'R 2 Vo dR/dt = a constant
(36)
The value of the constant is calculated from the value of the expression
at the surface of the pore, where R
density of the metal.
a, and
0-
,
the theoretical
So the velocity of the metal at R is
R = dR/dt
da)
=
=
7
a2
R2 g a dt
and the work dissipated in flow may be written, following Frenkel,
2
ik
ik
aR 2
2X
2
4
*Rd
2
a+ (da/dt) 2 -( -2 ) 2 yo dR/R 4
VO 2
(/
where Vi
is the flow tensor, 3
a (da/dt) 2
(38)
is the initial apparent density of the
porous metal, and V is the constant viscosity coefficient.
The work done
-
- 86
in contracting the surface of the pore under surface tension is
_ -a da/dt
-
(39)
(4Wa 2 )
(1dS/dt = - C
8
Equating the two expressions for work, we obtain
2 a (da/dt) 2 =-8'
(32/3)t(\%
S
da/dt
a da/dt
(40)
-
This gives us the initial rate of shrinkage of the pores of radius a.
At a later instant the radius of the pore is r, and the rate of shrinkage
is the density of the porous metal at that
of the pore is then, if
instant,2
dr/dt
-
3
= (da/dt) ()
A pore initially of radius a then has at any later time a radius
r
a +
0
($./st)
dt = a +
2a dt
(41)
dt
a - 2
Let
0
2
31
F
f 2TM
0
t
then from (41),
F
= (a/2c
)
-
(r/2 9-)
(42)
For a pore initially of radius 10-2 cm, then, a/2<s- = 4.166 x 10-6 if
we use the accepted value of c- = 1200 dynes per centimeter, and Table
(10) shows the progress of F as a function of r.
Similar tables could be
drawn up for other size pores, but they give a curve of the same shape,
for a pore starting with ten times the radius of another reaches a radius
one-tenth of its initial radius in ten times the value of F.
87
Table (10).
-
-
Radius of a pore initially 10-2 cm
as a function of F
r (cm)
F
10-3
3.750 x 10-6
10-4
4.125 x 10-6
10-8
4.167 x 10-6
5x10- 3
7x10- 3
8x10- 3
2.083 x 10-6
1.250 x 10-6
0.833 x 10-6
Figure (11) shows a plot of log F vs. log r for values of the initial
radius from 101 to 10-6 cm.
It is seen that for a considerable dis-
tance along the log F axis the curves are flat, then they curve sharply
downward towards radii of zero.
All during the flat part of the curve,
then, the density is very nearly constant, and is therefore not a function of t.
So for that region,
2
F
-
(43)
This permits us to find the time corresponding to the beginning of the
break in the curve, for the initial apparent density of the compact
0
is known, and the other factors are either constants or functions of
This break in the curve completely determines, then, the
temperature.
value of
if
the surface tension and the initial density are known.
Vacuum experiments designed to give a value of v
are reported in Part IV.
The difficulty of preventing surface oxidation makes these experiments
somewhat unreliable, and others reported there, involving various gas
pressures in the pores, give more exact results.
For comparison purposes the values of of Frenkel are calculated here.
~kT/D 5
predicted by the theory
According to him,
Q/DT
=kT e
/o
(43)
,4r;;L,
~
-L
1
+'4
"tFiPti44tZT
1tt-41 .
4-
I-'44-14
$~~~
H
+.I.
'--
-t1 ri'tIfr
Fi
E _
41-~
14
-4.*
4 ~7
1
ti
f
N
tt
I.4-t
~f i
4-j-
4
=F ii-
+
r4
L .j~~t
_T
-V!r
T i
-
V,~
IV7.
I ,
~-
I
4~ZT
9
t
_
"1I-.
L
1
4----
~~ -I
r,-r-
II
ti*
-tt
I
17
j
g
X+11
AA~
-- 1
I
~1
-,,..
~
where k is Boltzmann's constant, D is the self-diffusion constant, and
i is the lattice parameter.
The derivation of this equation has already
been given in the review of Frenkel' s work, (Part I, Section D).
For
copper the most reliable values of the constants involved (Hu5) are as
follows:
Lattice parameter 5
3.6 x 10~
cm
Coefficient Do = 11 cm 2/sec
Heat of Activation Q = 60000 cal/mol
Using the usual values (S15) of k = 1.38 x 10-16 ergs per degree, and
R = 1.987 calories per degree, we obtain
30000
T
1.28 x 10-16
11 x 3. x 10..g
T e
whence
0.544 - 10 t 13000/T + log T
log-NJ
Values of
are given in Table (11) below.
(45)
Figure (12) shows a plot of
the variation of the viscosity coefficient -\ with temperature.
Table (11).
Values of the viscosity coefficient - at
various temperatures
Temperature, OK
'emperature, oc
X\, sec/cm3
log v\
300
27
673
400
2.26x103 6
36.4
873
600
8
4.70x1012 2.43x10
12.7
8.4
1073
800
4.73x10 5
5.7
1273
1000
7.23x10 3
3.9
in a few minutes at 80000 takes a nearly infinite time at room temperature
to be precise, 4-3/4 x l030 times as long.
It is also evident why early
investigators found a "temperature of inception of sintering" that is
rather sharply defined.
For if sintering takes place in an hour at 8500C
-
From the table and the plot it is easily seen why a process which occurs
-90-
4
-
Lt
4
f-4.
H
4
-
IT.4
A
-~ -4
4L
~ ~
itt!~~~~
4
tt
L4.4
f~
-
j,
--------
~
tJ
14{~
*I -
f-.. .f4-
t-4jI
.
_.4
LT,.
7j~
4
'
71'
'r t
t
- 4~414:1
irir:oi.
T4
T4
4#
1
I
44,
14
..
'
1.
-{
4
4
rj~~
91
-
-
it takes nearly five hours to reach the same result at 8000C.
Furthermore
at any temperature the break in the curve of log F vs. log r-takes place
at different times, so that no density change of any consequence takes
place if the break is not reached during the time of sintering allowed at
is included in the time allotted, then extensive densification takes place
complete densification if a perfect vacuum were attainable.
B.
The Case of Gas Entrapment
If the pore initially contains a gas other than the vapor of the
metal, and that gas does not diffuse appreciably into the metal during the
time of sintering nor combines with it,
then we can set up the same equa-
tions relating log r and log F, taking account of the inside and outside
pressures; so if V is the volume of the pore and S its area, 'the work done
by the surface tension in contracting the pore against the pressure of the
gas is
P dV/dt -
dS/dt = 4P'Wa2 da/dt - 8-Wka da/dt
where P is the difference in pressure between the inside of the pore and
the outside of the compact.
Equating this work to the energy expended in
flow, as before, we have
32/3)
02 a (da/dt)
22
= (4P-ra2
-
ft a) da/dt
and
da/dt
3 x 4' a(Pa - 2
32W'a 3
)
02
=
(Pa - 2
8N
)
(46)
3 2
The difference in pressure P is related to the absolute pressure PO which
exists at time t n 0 inside the pore and the external pressure P1 by
-
any one temperature, but if the temperature is increased until the break
-
- 92
PO a3/r3
P + Po = po 3
3Wr
3
At any instant after the beginning, then,
dr/dt
=
(Pa - cr
2-r)
2
(7)
and, integrating,
r
dr
4
8A
2
aoa-5 r
2 d
d3
F
18
(18
0fo
If the compact is either compressed or heated in the presence of a gas
until the pressure in the pores is Po, and is then heated in vacuo, where
PI
0, then the left hand integral of (47) yields, letting Poa3/r2
.then r
dr
-
X2-;
dx, and
x2
-Pa3
(4iZ
r2
dx
x(2
2
-
4
Pa3
o 2
joa
-
( x
+
2;
.n
2e
x
T2Q
- 2
x+
F
(48)
Points on the curves of log F vs. log r according to Equation
(48) are calculated in Appendix V for values of Po of 100 dynes per square
centimeter (a good vacuum of about 1
), 103 dynes per square centimeter
(1000), 106 (one atmosphere), and 109 (7j tons per square inch, or of
the order of magnitude of the gas entrapped in pressed copper compacts).
e-2
o
93
-
-
The results are plotted in Figures (13), (14), (15), and (16).
several features of interest.
They show
All the pores begin to shrink and complete
their shrinkage within a restricted length of time, which depends only on
the pressure Po initially inside them.
Smaller pores begin to shrink and
complete their shrinkage before larger pores have begun.
All pores reach
an equilibrium radius which stays constant presumably forever unless the
gas subsequently diffuses out of them.
The ratio of initial to final
radius is greater for smaller pores, so that for equal entrapped gas pressure finer pores will shrink more than larger ones.
For higher pressures
of entrapped gas the finer pores shrink but the larger ones expand; furthermore the shrinkage of the finer pores takes place before the large pores
have begun their expansion.
The application of these results to practical powder metallurgy
is restricted to the vacuum sintering of pressed compacts.
The more general
case of sintering of pressed compacts in a gas or of unpressed compacts
in a gas requires a much more elaborate calculation which is carried out
below only for pores 10-2 centimeters in radius and for an external pressure of one atmosphere (10 dynes per square centimeter).
But the result
of the restricted solution is very close to that of the general solution
in the case of high internal pressures, where the external pressure makes
little or no difference.
In other words, it is to be expected that Figure
(16), giving the sintering curve of compacts with 7j tons per square inch
of gas pressure inside, is independent of the outside pressure up to at
least several atmospheres.
And in Drapeauts work on compressed copper com-
pacts the results are seen to be in agreement with the curves of Figure (16).
One of the diagrams drawn by Drapeau (Dr2) is reproduced here (Figure (17)).
4--7.
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Figure (17). Sintering curves for compressed copper at various
temperatures. Note shrinkage preceding expansion and time-temperature
correspsdence.
The ordinate in the change in volume for copper cylinders sintered at
various temperatures for times shown on the abscissa.
The compacts first
shrink -- the finer pores are disappearing, as is predicted by Figure (16)
and then expand -- the larger pores are becoming active, and, as shown in
Figure (16), these expand rather than shrink.
Furthermore, it
is seen
that at higher temperatures the whole time cycle is shortened without
changing the shape of the curves, in accordance with the calculation of
the function F, which is proportional to time and varies exponentially
with temperature.
The matter of the exact dependence of F on temperature
will be considered again below.
Drapeau's results do not lend themselves
to exact calculation because it is impossible to predict the pore size
in his compacts from the data available, but qualitatively they are in
agreement with the results plotted here in Figure (16).
Similar qualitative confirmation of the curves calculated here
--
-
- 99
is found in the publication of Trzebiatowski (Tr4) in which he describes
the course of densification of copper compacts pressed in air and then
sintered in vacuum.
His results have been shown in Figure (4), page 24.
The abscissa here is the temperature, the ordinate the density.
With the
introduction of a logarithmic scale for the abscissa the latter may equally
Then the same phenomena may be observed
well represent the function F.
in the highly pressed compact the inter-
as were found in Drapeau's work:
nal pressure is high, and therefore according to Figure (16) the compacts
first shrink, then swell up.
At the lower compaction the internal pressure
is less, as in Figure (15), for example, and therefore the compacts shrink
continuously, but tend to reach a constant density differing from the theoretical density of copper.
The next case to be considered is that in which gas is entrapped
in the pores of the compact, but it is a gas which forms a compound with
the metal.
In this case the pressure of gas which is in equilibrium with
the compound at the temperature of sintering is constant.
Assuming that
the compound formed does not dissolve in the metal or in any other way
affect the surface tension appreciably, we can find the change in pore size
as before from the equations of work and energy.
between the equilibrium pressure P
and that outside the compact, PO, dur-
ing sintering, then
1
(Pr - 2 )
dr/dt
j1r
where P = Pe - Po
.
far
For if P is the difference
r -
8
f
Y(P dt
(49)
F
Let Pr
-
2
100
-
-
= x, then dx/Po = dr, and when r
= a, x
= Poa - 24
Pr-2r
1x
F
P
1x
O jPa-2
X
-.3 log Pr -2r
PO
F
(50)
Pa - 24
To follow the change in radius of pores of various initial radii, a, the
values of log F for various values of log r and a are calculated in
Appendix VI.
below.
The results are plotted on Figures (18), (19), (20), (21)
It is seen that the results do not differ very much from those of
Figure (11), at least for small pores, and it is only when the equilibrium
pressure of the gas is very high that the figures show divergence from the
gas-free case.
silibities.
it
The application of this calculation shows interesting pos-
As will be shown in the description of experiments that follows,
is extremely difficult to attain the conditions of Figure (11), or even
of Figures (13) and (14), that is, to remove from the pores most of the
gas.
The reason for this is that at the temperature at which the gases
are being evolved most rapidly (around 4000C for copper) sintering is also
going on rapidly, and the pores are closed up before all the gas can be
removed even by a high-speed vacuum pump.
In order to obtain completely
the
dense compacts, therefore, it appears possible to proceed as follows:
compacts of copper powder are compressed not in air, which contains neutral
nitrogen, but in pure oxygen; upon subsequent vacuum heat-treatment the
oxygen entrapped in the pores combines with the copper, leaving a low pres-
sure difference between the gas in the pore and the vacuum outside.
If the
copper powder is initially carefully deoxygenated the oxygen in the pores
might not interfere too much with the initial adhesion of the particles.
4
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-
- 105
The calculation carried out above for pores in which there is
a gas capable of forming a compound with the metal must be applied with
discretion, because whenever the gas pressure is lower than the equilibrium pressure no compound is being formed, and the problem is that of
Figures (13) to (16), in which a gas is present which does not react with
the metal.
Closely allied with the case of compound formation is that of
a pore gas which diffuses very rapidly with respect to the time of sintering.
In this case the outside of the compact would sinter in accordance with
Figure (11), as if there were no gas present, but on penetrating deeper into
the compact one would find the pressure increasing as predicted by the diffusion equation, and the central material would successively pass through
the processes shown in Figures (21), (20), (19), and (18).
C.
The General Case
Equation (47) has been solved above for the special case of heat-
treating in vacuo, that is, for P10 : 0.
the general solution of the equation.
Of considerable interest also is
It represents the heat-treatment
in gas of compacts with gas entrapped in the pores.
The solution is quite
complicated and involves the solution of two auxiliary equations, determined
differently in different individual cases.
fore prohibitive.
The labor of solution is there-
The method of solution is shown in Appendix VII, where
the equation is solved for pores 10-2 centimeters in radius for the particular case in which P
= 106 dynes per square centimeter and P1 is also 106
dynes per square centimeter.
These figures represent the condition found
in the sintering of unpressed powder in a neutral gas at a pressure of one
atmosphere.
The result of the solution shown in Appendix VII is the plot
-
- 106
of Figure (22), which shows in addition sections of the curves of Figure
(11) and of Figures (13) to (16) relating to pores of initial radius 10-2
centimeters.
The curves show that the time of rapid sintering comes at
the same time as it does for other conditions, but that there is no swelling as is the case in vacuum sintering with gas of pressure 106 dynes per
square centimeter entrapped, and yet the stable pore radius attained is
much greater than in the case of vacuum sintering with lower pressures of
entrapped gas.,
As will be shown in the next part, where the experimental work
is presented, cycles of sintering may be developed in which the compact
is made to follow one curve after the other, within the limits imposed by
the material used.
Thus, compacts are sintered first in argon at one at-
mosphere, following the middle curve of Figure (22), and then are sintered
in vacuo, so that they follow a process close to that represented by
Figure (15), and expand again.
That such a procedure is possible is clear
evidence that gas entrapment is responsible for swelling of compacts, and
is also evidence in support of the theory of sintering presented above, in
which changes occur by flow through the body of the metal rather than by
surface diffusion, recrystallization, or any other of the host of other
hypotheses presented in the literature; (see Introduction).
D.
The Influence of Errors in the Constants for the Metal
In the preceding sections the abscissa of all the curves has been
represented in terms of the function
t
2
F 8Y\ ?'Jo 1i dt
This is a function both of time and of temperature, for -qisstrongly
4~
Lrt~1-;
0-rr
_.4
141
-4----
t4TI
J
~
.X.
T$~
177-~
4
t
a,1
1t
L
J:5
._4~
44
-r77rt7
A
Ti
..........
T144
167-
t
z
-T-
T
1'
1
4 4;
74
t
#+
4-
4i
J
j
1 - -!1.. 9
:
~4
0-
.- A.
4;1.
+
108
-
-
dependent on temperature, as shown above in Section A.
If we now take the natural logarithm of this equation
in F = ln (
2)j
dt) -3ln
(51)
and differentiate, we obtain
dF/F =
pi at
JF t
a
-
dy/'
(52)
But we can write, as shown above in Part II, Section C,
Q/RT
kT
kT
Yt= g
08g
(53)
e
so that, taking the natural logarithm,
ln q
: ln k + ln T - ln
Do - ln
i.Q/RT
(54)
and differentiating,
d 3J/since k, R, and
5
:
-
dDO/0
+ dQ/RT
(55)
have been accurately determined and the control of T
in modern furnaces is as good as one wishes to make it.
Equation (55)
then states that the relative error in Y is the sum of the relative error
in Do, the temperature-independent term in the constant of self-diffusion,
and the term dQAT, which is not the relative error in the heat of activation of diffusion, but the actual error divided by RT.
given errors in D0 , in the density
Jy,
and in the heat of activation of
self-diffusion Q can affect the value of F.
Equations (52) and (55) may
be combined to yield
P't2
dF/F
Let us see now how
to lp t dD0 d-dQ/RT
-i
dt
(56)
109
-
-
In the calculation of the position of the break in the curves of log F
independent of time and equal to the initial
vs. log r we may consider Pf
density fo.
Then Equation (56)
becomes
dF/F - dD /Do0 - dQ/RT
(57)
since the time may be measured as closely as one wishes, so that dt, like
dT, dk, dR, and d S previously, is equal to zero, or is negligibly small.
Equation (57)
can be solved for values of errors in Do and Q to
At 850 0C the value of RT is 2245
give the corresponding error in F.
The value of Q for copper is given in the literature variously (&atIT)
calories.
from 57200 calories per mole to 61400 calories per mole, a spread of 4200
calories or 7j per cent.
The error in F is then
dF/F
Values of D
-
4200/2245 x 100 = 187%
are given usually (Hu5) with a probable error of 5%.
corresponding error in F is then also 5%.
The
It is seen in the calculations
that precede this section that the surface tension <r enters into all the
equations in the first power.
An error in this factor accordingly leads
to an equal relative error in F.
It has been shown in Part II, B, that
the value of 1200 dynes per centimeter for copper is probably good to within 100 dynes per centimeter.
This is an error of about 8%.
The total proba-
ble error in F due to these three factors amounts, then, to the sum of
these three figures, namely 200%.
At lower temperatures than 85000 the
value of RT is less, and therefore the error introduced into F by an error
in Q is even greater.
The value determined for the heat of activation
of self-diffusion is therefore of paramount importance in solving the question of the mechanism of sintering.
Since y is the only temperature-dependent factor in our equations,
-
- 110
the error in its value may also be considered as an uncertainty in the
temperature for which the densification curves are calculated.
In summary of the results of these calculations, it has been
shown that relatively small errors in the value of the heat of activation
of self-diffusion introduces very great errors in the parameter F, the
function of time and temperature.
It
may be also pointed out that the sur-
face tension always enters into the equations of sintering in conjunction
with a gas pressure term.
Therefore, it is to be expected that experi-
mental curves showing the shrinkage of a pore containing entrapped gas
will not necessarily correspond to better than 8% with the appropriate
curves calculated above.
E.
The Correspondence between Time and Temperature in Sintering
The parameter F is a function both of time and of temperature.
Since the sintering curves are uniquely determined for any particular
value of F, it follows that there is an analytical relationship between the
time of sintering at any one temperature, ,and the temperature of sintering
at any one time.
The results of Drapeau mentioned above (Dr2) can be used to
check this relationship.
For on Figure (17), taken from his paper, the
curves are seen to cross the lines parallel to the time axis corresponding
to a given change in volume at various times of sintering, for various
temperatures of sintering.
If the abscissa had been in terms of F rather
than of time, the theory indicates that all the curves should be congruent
and therefore should cross lines of equal volume change at the same F value.
Therefore if T and T' are two temperatures, the times t and t' at which
the curves cross any abscissa are related as follows:
Yt
pf>
111
-
-
8 -F
kT
and therefore
t
T' eQ/RT
T~ Q/RT
(58)
t
Taking logarithms,
log (t/t) r log Tt/T -
2.3R
(T,
-
T
)
(59)
In Table (12) are presented some of Drapeau's data taken from the figures
in reference Dr 2, and calculated to Equation (59).
Table. (12).
Data from Drapeau, and Correspondence between
Time and Temperature of Sintering.
Fig. No. (Dr2)
%
Vol change
t (#en)
t' (itea)
log t'/t
T' OK
T
OK
1Log TI/T
1
0
160 150
580 550
.56 .56
1088 1088
1200 1200
(Q/2.'3R)(1/T' -/T)
sum.
153 153
-. 04 -. 04
1.11 1.11
1.07 1.07
153
-1
153
-2
153
-1
17
30
45 300
30
.33 .42 1.0
978 978 857
1088 1088 978
-. 05 -. 05 -. 06
1.35 1.35 1.88
14
154 154 154
5
4
3
140
110
600
.63
520
.67
102
450
.64
1088 1088 1088
1200 1200 1200
-. 04 -. 04 -. 04
1.11 1.11 1.11
1.30 1.30 1.82 1.07 1.07 1.07
154 154
1
2
96
90
380 320
.60 .55
1088 1088
1200 1200
-.04 -. 04
1.11 1.11
1.07 1.07
The table shows that the results are not consistently in agreement.
Either
the value of the heat of activation (60000 calories per mol) has been
chosen too high by some 20000 calories, or else the times reported by
Drapeau either include the heating up period or are furnace times, the
compacts having possibly not heated up as fast as the furnace.
The latter
explanation is preferred, as the discrepancy becomes less at longer times,
and if all the time values are shortened by some sixty m&iAts, the results
are in fair agreement with the calculation.
Part IV has dealt with the calculation of curves of sintering:
-
- 112
plots of the relationship between the radius of a pore and the parameter
F, which is a function of time and of temperature.
It is shown that in
the absence of gas in or out of the porous metal the pores all tend to
disappear, the finer ones going through their disappearance process before
the large pores have begun to shrink at an appreciable rate.
This is in
agreement with the summary of Rhines, in which he states that small pores
seem to shrink faster than large ones.
One may conclude that powders made
of particles all of one size begin to sinter at a more definite time, sin-
ter more rapidly once they have begun, and reach their final density at a
more definite time than powders made up of a range of sizes.
In agreement with Rhines (Rhl), who states that finer powders
sinter more rapidly than coarser powders, it
is found here that finer pores
sinter more rapidly than coarser pores.
Also in agreement with Rhines it
is found that the rate of sin-
tering decreases with time, and in the case of gas entrapment the density
reaches a steady value.
If the compact has been pressed in such a way as
to entrap gas at high pressures, the compact, instead of shrinking, expands,
but if it is made of finer powders the tendency to expand is less.
Pores
in expanding tend to become spherical, so that if a compact is pressed from
one or both ends rather than by hydrostatic pressure, pores which were
originally isometric are pressed flat, and on sintering expansion will take
place predominantly in the direction of pressing.
If at the same time
there are smaller pores in the compact, these do not expand but shrink,
and it
is therefore possible to have expansion in the direction of pressing
occurring simultaneously with an overall shrinkage which shows itself as
a net shrinkage perpendicularly to the direction of pressing.
-
- 113
The curves show that a compact sintered with entrapment of gas
may then be sintered again at a lower outside pressure and the reverse
of sintering may then be accomplished: pores are enlarged again after
having once been shrunk.
IV.
Experimental Results
A.
Sintering without Entrapment of Gas
The experiments were designed to answer the following questions:
1.
Does the process of sintering follow, at a given temperature,
the time curves calculated in Part III?
2.
does it
If
so, what is the heat of activation of the process and how
compare with the heat of activation of self-diffusion?
3.
What can be observed concerning the recrystallization in con-
nection with the presence of the boundaries of the original particles?
7.
What can be predicted as to methods of obtaining better sinter-
ing in a shorter time or more effectively controlled sintering?
In order to interpret the results most convincingly it was
necessary to select a material which could be easily packed without pres-
sure into an array containing pores of uniform size.
were therefore selected.
Spherical particles
By the atomization process it is possible to
produce spheres of very good geometrical figure and of very uniform size.
The particle size chosen was coarse enough to allow a good packing to be
made, and yet not so coarse as to prolong indefinitely the times required
for sintering.
Powders were therefore examined and two types were chosen
in a firstealection.
These were ordered, and when they arrived it was found
that one of them was far superior to the other in regard to the properties
required.
This was furnished free of cost by Macleer Manufacturing Co.,
114
-
-
Greenback, Tennessee, for whose kind cooperation the authorts thanks are
here tendered.
The specification is:
Atomized Copper Powder, Electrolytic
Copper (purity (99.90%), all minus 100, plus 200 mesh, U. S. S. screens,
Order No. B-21677, Sample No. 1.
This powder was found to be made up of
very good spheres containing a very small proportion of fines in the form
of finer spheres and irregular particles.
It was subjected to a further
sieving process, using standard Tyler sieves and a Ro-Tap machine, through
sieves Nos. 100, 140, 200, and 325.
Most of the powder was collected in
the 100-140 mesh range, and this part only was used in the experiments,
except for a few heats in which finer fractions were sintered alongside
the 100-140 mesh fraction to check. qualitatively the influence of decreasing the pore size.
The sieving on the Ro-Tap machine lasted one hour.
A
check sample showed that sieving for a longer period gave no further appreciable separation.
The measurement of the size of particles was made
microscopically on the plate of Figure (23), which is reproduced here.
Seventy-five diameters were measured and averaged, and gave an apparent
dimension of 0.377 inches on the polished section at a magnification of 100.
If the plane of polish is considered parallel to the equator of the particle, then the probability that the plane of polish has cut through the
sphere at any latitude is equal to the sine of the latitude.
Since the
plane of polish cuts the spheres indiscriminately, it follows that the average latitude at which the plane of polish has cut the spheres is 30 degrees,
where the sine of the latitude is 1/2.
The true diameter of the particles
is therefore
d = do/cos 300 = 0.377/0.866 = 0.435"
at 100 magnifications.
This is equivalent to a radius of 0.011 centimeters.
-114a-
4,4
Figur. p3). Specimen heated in vacuo 45 hOurs. Magnification
10M. 50UH4OH-50%H2O2 etch.
-
- 115
To contain the powder specimens, sheet copper 1/8" thick was
cold-rolled to 0.053", annealed 2 hours at 4500C, quenched in water from
'
that temperature to remove the scale, and stamped out into discs l
diameter.
These discs were then deep-drawn in a die mounted on a Tinius
Olsen. Universal Beam Type Testing machine into little cups 15/16
inside
diameter, 3/4" deep, with a tapering wall so that the compacts could be
removed easily.
They were finished on a lathe and coated on the inside
with a suspension of kaolin in shellac dissolved in methyl alcohol; the
crucibles were dried and fired to drive off the volatile matter in the
coating.
This produced an adherent coating which prevents adhesion of the
copper powder to the copper crucible.
The crucibles were charged with the powder as required, with no
pretreatment since all specimens were to be treated with hydrogen in the
furnace immediately before treatment at high temperatures.
The first series of heats were performed in a furnace constructed
as follows:
A quartz tube le' inside diameter and 24" long extended 12"
into a molybdenum-wound furnace.
The portion of the quartz tube that was
outside the furnace was surrounded by a steel tube cooled by means of a
copper coil through which water was run.
It was expected that the expul-
sion of the gases from the metal would take place mostly at about 40000,
where the sintering curves show an appreciable rate of sintering.
It was
therefore imperative that a very fast vacuum system be used in order to get
the expelled gas out of the pores before the latter became closed off by
sintering.
For this reason the quartz tube was connected directly by means
of a glyptal-coated rubber connection to a Genco Hypervac model 100,
which has a capacity of 15000 cubic centimeters per second in free air,
116
-
-
-S5
and a guaranteed vacuum of 10 millimeters.
The diameter of the smallest
passage for the gases being pumped out was 1" over a length of
3 ".
As
will be shown in the data, even this system was unable to remove the gas
from the pores before they became closed up.
At first the specimens were
heated at 45000 for 2 hours in hydrogen to deoxidize the particle surface
and to get as much as possible of the oxygen out that is dissolved in the
metal. -Then the hydrogen connections were closed off and sealed with
glyptal varnish, and the vacuum was turned on.
was rapidly heated to 85000.
kept at 20 minutes.
Simultaneously the furnace
The heating up time from 4500C to 8500C was
Temperature control was obtained by means of a
Model 214D Capacitrol B made by Wheelco Instrument Co., Chicago, Illinois.
The thermocouple was inside the quartz tube, and separated from the crucibles
only by a l/l6" steel strip on which the crucibles rested.
The thermo-
elements were chromel and alumel, thermocouple grade, and were of 20 gage
wire.
The temperature at which the Capacitrol was set was checked repeatedly
during the run by means of a Brown Portable Potentiometer Model 1117, which
contains its own standard cell.
At the end of the run the quartz tube was
withdrawn from the furnace and held in the water-cooled zone.
To hasten
cooling hydrogen was introduced after the temperature had dropped to about
50000.
The vacuum was measured by means of a Central Scientific Company
thermocouple vacuum gage.
It was calibrated against a McLeod gage in the
laboratory of Professor Stockbarger by the writer with the kind assistance
of Mr. Williams of that laboratory.
The pump used in that laboratory pro-
duced a vacuum of less than 10-5 millimeters, but at that level the McLeod
gage was not accurate.
Consequently measurements were taken only down to
5 x
io5
millimeters.
117
-
-
The vacuum obtained in the furnace during the runs
was considerably better than this figure, and extrapolation of the calibration curve indicates that it was probably better than 105 millimeters.
The calibration curve is not recorded here since it is only valid for
much lower vacua than those used.
The first results of heats in vacuo at long sintering times showed
that a stable density was being reached after about 15 hours, and consequently that residual hydrogen at a considerable pressure remained in the
pores even after the treatment described above.
This series was completed,
but will be reported in the next section, dealing with sintering involving
gas entrapment.
Another attempt was made to obtain vacuum conditions inside the
compacts.
one-hal
This time the charges were heated up in hydrogen to 4500C in
hour, held at 4500C for only five minutes, and the vacuum and
heating current then turned on.
This system may have solved the difficulty
of the hydrogen entrapped in the pores, but it introduced a worse one.
For, although for short sintering runs the results were excellent, for longtime runs the vacuum of 10-5 millimeters was not sufficient to prevent considerable oxidation of the material.
While this would not have prevented
adequate measurements of density if the oxidation were confined to the
surface of the compacts, the time and temperature conditions were favorable
to considerable oxygen diffusion into the compact, probably in the form of
copper oxide going into solution into the metal.
As a result, the compacts
in this series also failed to represent pure vacuum conditiona.
In the
case of very long runs oxidation was so bad that the compacts were dis!carded as worthless.
U8
-
-
It is probable that in the first series some oxygen entered the
metal but due to the presence of entrapped hydrogen it was immediately converted to water vapor, which added to the gas pressure inside the pores.
This matter is discussed more fully in the next part, in which the results
are discussed in the light of the theory.
Attempts to arrive at conditions of pure vacuum were then abandoned, since they would require the installation of elaborate static systems,
and even with those there was no guarantee that the conditions would be
improved.
In fact, it is extremely improbable, as will be shown in the
discussion, that gasless pores are ever attained in practice, and only by
entrapping pure oxygen or some other gas which will form a compound with
the metal can conditions approaching those of Figure (11) be attained.
And in that case other difficulties are introduced, and the value of the
surface tension is no longer that reported in the literature.
B.
Sintering with Gas Entrapnent
Two series are included in this heading.
The first is the series
mentioned above in which vacuum conditions were not attained.
The second
is a series in which the crucibles containing the copper powder were heated
in argon containing a very slight amount of hydrogen.
sure was one atmosphere.
The total gas pres-
The proportion of hydrogen in the argon was meas-
ured by means of constriction flowmeters calibrated in the laboratory previously by the writer and two of his colleagues in connection with the work
of one (Pr/4) of them.
on the flowmeter.
The proportion of hydrogen to argon is 1.3 to 15.0
This corresponds to 8% hydrogen in a flow rate of 33
cubic centimers per second.
The hydrogen was purified by passing it over copper gauze at
119
-
-
85000, then over activated alumina, then over palladium alundum, a catalyst
for the conversion of residual oxygen to water vapor in the cold in the
presence of hydrogen, and finally over indicating drierite.
The argon was
dewatered over indicating drierite, and then passed over magnesium turnings held at 50000 to remove the residual nitrogen and oxygen.
The argon
used was lamp grade, containing 99.S% or more of argon.
The furnace used for this series was initAally the same as that
described in the previous section for the vacuum experiments, the vacuum
pump being used only in a preliminary degassing of the powder at about
10000.
The heating up time was 45 minutes to 45000, and then the current
was increased to bring the remainder of the heating up period up to 400
degress in 15 minutes.
The total heating up period was therefore one hour.
In the middle of this series the molybdenum furnace was burned out due to
failure of the thermocouple, and a furnace was wound with Chromel A 20 gage
wire to fit the quarts tube.
Table (13) gives the data obtained in the two series.
The first
column is the time in hours from the moment the compact reached 850 0G.
second column gives the density of the resultant compacts.
The
This was meas-
ured in the conventional way by suspending on a fine wire, weighing in air
and in water (with a small quantity of Duponol detergent), and computing
the density from the difference in weights. The specimens were boiled in
wax to prevent water from seeping into the pores during the weighing.
The
weights were duly corrected in the usual manner for the weight of the wire
in air and in water, the weight of the wax, and the density of the water
at the temperature of the measurements.
The weight in air was not corrected
to vacuum. The third column gives the density as a fraction of theoretical
120
-
-
density for copper, taken as 8.99, as calculated in Part I, Section B,
from lattice parameter and atomic weight.
The fourth column gives the
fraction of the volume occupied by pores.
This is the complementary frac-
tion to that of column three, since the volume of metal plus the volume
of voids makes up the volume of the compact.
If the spheres had been perfectly packed in cubic array, the
volume fraction occupied by metal would have been the ratio of the volumes
of a sphere to that of a cube of edge equal to the sphere diameter, or
'T /6 :0.52.
The starting density of the powder used in the experiments
was the fraction 0.53 of theoretical density.
It may be therefore assumed
that the packing in general was not far from perfect, although it assumes
a packing that is not geometrically perfect, and therefore includes arrays
that are denser than the simple cubic as well as regions where the array
is less dense.
It is consequently reasonable to assume that the number of
pores is the same per particle as that existing in a perfect array, cubic
or otherwise:
all have evidently one pore per particle.
On this basis,
and using the average particle radius calculated in the last section from
measurement of Figure (23), the volumes of the pores is calculated from
the fraction of void space and the number of particles to be those reported in column five.
The corresponding radius of the equivoluminous
spherical pore is given in the sixth column.
rithm of the time.
The last column is the loga-
-
- 121
Table (13). Densities and Pore Volumes and Radii Obtained
in Sintering Experiments on Powder of Particle Size
1.1 x 10-2 cm (radius)
I.
Experiments in Vacuum
Time
Density
(hro.) gr/cm 3
0
4.740
1
4.738
2
4.770
4
4.653
6
5.172
8
4.652
11.5 5.004
15
6.322
45
6.167
105
6.052
II.
0
7
12
13
16
20
As Fraction
of 8.99 g/ce
0.527
0.527
0.531
0.518
0.575
0.517
0.557
0.703
0.686
0.673
Pore
Fraction
Pore o1ome
x 10cm?
0.473
0.473
0.469
0.482
0.425
0.483
0.443
0.296
0.314
0.327
5.00
5.00
4.96
5.10
4.49
5.10
4.68
3.13
3.32
3.46
1.06
1.06
1.06
1.07
1.03
1.07
1.04
0.91
0.93
0.94
0.00
0.30
0.60
0.78
0.90
1.06
1.18
1.65
2.02
0.473
0.439
0.403
0.338
0.337
0.352
5.00
4.64
4.26
3.57
3.56
3.72
1.06
1.03
1.00
0.95
0.95
0.96
--0.85
1.08
1.11
1.20
1.30
Pore -adius
x 14cm
Log Time
Experiments in Argon
4.740
5.044
5.368
5.949
5.957
5.826
0.527
0.561
0.597
0.662
0.663
0.648
1
The data of Table (13) are plotted on Figures (24) and (25).
The specimen heated for 6 hours in vacuo was mostly blown out of the
crucible at the beginning of the experiment when the hypervac 100 pump
was turned on without previously closing off the tube used for hydrogen
exit.
Not knowing the extent of the damage, the heat was carried out to
completion as planned.
It would normally have been discarded and repeated,
but in this case it is included in the data for this reason:
the only
difference between it and the other compacts is that it was much smaller
as a result of being mostly blown away.
It therefore represents a compact
in which the residual hydrogen pressure is probably less than in the other
compacts, and so it approaches more closely the conditions of Figure (1),
122
-
-
i.e. it has begun to shrink at a slightly earlier time than the compacts
with more gas pressure inside.
Both curves show the characteristic break in the sintering curve,
and the flattening out after the establishment of a stable pore radius.
Both curves also indicate a slight rise after the main fall in the curve.
This indicates that the packing was not perfect, and that there are a slight
number of larger pores in the array; these larger pores, according to the
calculated curves, expand rather than shrink, and their expansion begins
after the finer pores have begun to shrink.
This rise is therefore addi-
tional check on the theory.
Qualitatively, both curves are in accord with the results of the
theory.
Let us see if the same accord is present quantitatively.
The
initial radius of the pores is 1.06 x 10-2 cm. From the middle curve of
Figure (22) we see that the value of log F for which the change in radius
has reached about one-half of its course is about -6.5, or F = 3 x 107.
The corresponding viscosity coefficient is then
2
8
Pa
Vpz
t ,
3x10-7
where the time t is in hours.
8
0.62
x 3600 x 107. t = 1.73 X 109t
(60)
In our experiments the break in the curve
is about half completed at 12j hours, so the corresponding value of the
viscosity coefficient is
: 12.5 x 1.73 x 109
=
2.16 x 1010
From Equations (43) and (43a) the value of Q corresponding to this value
of the coefficient at 8500C is
Q
=
(9.456 - 3.050 + 10.334) x 5166 = 864?9 calories per mol.
This is considerably higher than the heat of activation for self-diffusion
(60000 calories per mol).
The time for the break in the curve for argon
123
-
-
and for vacuum sintering is the same within the accuracy of the experiments,
so that this value of heat of activation applies to both series.
In the argon series the change in pore radius is from 1.06 x 102
to about 0.95 x l-2O, a fall in the log r value from -1.975 to -2.033, or
-0.058.
The difference predicted by the middle curve of Figure (22) is
-0.034.
If the theory is correct, then the larger value of the radius
change indicates that some shrinkage takes place before all the pores are
completely closed, or else the pores are slightly smaller than calculated.
Since the rise in the curve indicates the existence of larger pores than
the average, it follows that the average size of the pores contributing
to the main break is less than that calculated for perfect packing.
The
discrepancy in the values is therefore in the right direction, and the
explanation of a smaller pore size than that calculated is the one preferred.
The series sintered in vacuum shows a more considerable change
in radius, from 1.06 to 0.92, or a fall in the logarithm of r from -1.975
to -2.036, or a difference of -0.037.
From curves in Figures (14) and
(15) this is equivalent to a pressure of about 150 mm. inside the pores
at the time of closing.
-
I li
I
-
-
T
I
Tr
44
<4
4
~j.
-7
T ~1
t nT
Ti
-
.1r
i
-H
-
771-77'
I
-44
-,
_1
7
-r-
___
OEO
I
100
S
TIT
F
4
7--
r1ThTTt~
T4F
I 4-
-H1H
BIL
7
71
I
7,,
7 -T
-"44
-4
.-t-tt
It
4
46.
77 71 1 j - - , , ,
A-
-17
A
-- 7
jt.
Vir
+4
4-+
T-,
1rM
-
- 126
C.
Influence of Pore Size
The theory predicts that if the pores are smaller they shrink
more rapidly.
Accordingly some specimens each of the -140 + 200 and -200
fractions of powder were heated in argon as described above.
and Figure (26) give the results.
Table (14)
The curve for the -100V140 series is
also shown in the Figures.
Table (14). Densities and Pore Volumes and Radii Obtained
in Sintering Experiments in Argon at One Atmosphere on
Powder of Various Particle Sizes.
I.
-1404200 mesh
Time
Density
(Hrs.) gr/cc
0
7
12
13
20
II.
0
7
12
13
As fraction
of 8.99gr/cc
5.079
5.177
5.607
5.880
5.865
Pore
Pore Vol e
x 100 cm'
Fraction
Pore _adius
x l1F cm
Log Time
0.565
0.576
0.624
0.654
0.652
0.435
0.424
0.376
0.346
0.348
2.20
2.14
1.90
1.75
1.76
0.81
0.80
0.77
0.75
0.75
0.85
1.08
1.11
1.30
0.556
0.619
0.639
0.688
0.444
0.381
0.361
0.312
1.30
1.12
1.06
0.92
0.68
0.64
0.63
0.60
0.85
1.08
1.11
-200 mesh
4.998
5.562
5.743
6.183
The results qualitatively confirm the calculations since the break in the
curve occurs a little sooner and is a little deeper for finer pores.
Unfor-
tunately the starting radii of the pores are not very different for the
three powder sizes.
in the vicinity of
As can be seen in the Table, all the pores are still
0-2 centimeters in radius.
As expected, due to the
fact that the pores are smaller, fewer of them expand, and the rise at the
end of the curve is not so great for finer powder.
Also as expected, there
are many more finer pores than the average in the statistically wider ranges
TJII
-I
tt~
-H
4-
- -H
-
-
-t-
+
-
--
-
- -
" LT
-4-
E44+
-
TT
7
tt
!7-T-
2---------
r.0
5
0
-J
128
-
-
of sizes represented by the lower sieve fractions, and consequently their
effect in depressing the curves at the beginning is more marked than it
is for the coarser and more uniformly sized fraction between 100 and 140
mesh.
D.
Influence of Temerature on Heat of Activation
One run was made to check the value of the heat of activation
at a higher temperature.
The two top fractions of particle size were
used, and these were heated in argon as described above for two hours at
9000 C.
Other runs were not made because the time of heating-up at this
temperature is of the order of magnitude of the time for the break to
appear in the curves.
Shorter heating-up times were not indicated because
the results of Huttig, mentioned widely in the introduction, indicate that
all the gases are not evolved in much less than two hours at 4500C.
Heating
up from that temperature to 9000 C cannot be done in much less than the 15
minutes used in all the experiments.
A lower temperature than 8500C could
not be used, for the break in the curve would not then appear until some
60 or 70 hours of sintering time.
As expected from the calculations made above (Table (11)) the
time for the break in the curve at 9000C is to be expected at about 1/5th
the time required at 8500C, or a little over 2 hours.
The two experiments
show that at two hours the break is beginning, for the density has increased
from 4.740 to 5.254 for the larger particle size fraction and from 5.044
to 5.347 for the middle size fraction.
(26).
The points are plotted on Figure
E.
129
-
-
MetallogravhT
The metallography of sintered porous compacts such as were
prepared in these experiments offers some features of interest.
If the
usual procedure is used of rubbing down on emery cloths of decreasing
grit size, then of polishing on a felt or velvet lap with levigated alumina,
and finally of etching in the standard etchant for copper, namely 50%
ammonium hydroxide and 50% hydrogen peroxide, the result under the microscope is a field such as that shown in Figure (214.
This photomicrograph
is taken at a magnification of 10OX on a Bausch and Lomb Metallograph,
Research Model, with bright field illumination.
The objective was 8X; the
ocular was a 5X Huighenian; the bellows extension was 59j centimeters.
The plate was a Wratten Metallographic Plate.
The specimen shown in
Figure (2) is one of the series heated in vacuum, and gave a density of
6.052, or a porosity of 32.7%.
The photomicrograph shows no porosity at
all. Figure (274) shows the same specimen after repolishing as follows:
The original polished section was considerably overetched, then ground
down again on 3-0 paper, 4-0 paper, then on the canvas wheel, using levigated alumina (solution #1), then on the second canvas wheel, using a finer
suspension, then on the finest suspension, using a velveteen base.
was then again etched normally and examined for porosity.
It
If this porosity
differed from the previous examination the entire process was repeated
until the porosity seemed to be constant.
As can be seen, the pore dimen-
sions vary considerably but the assumption of one pore per particle is
reasonably well fulfilled. A great deal of the variation in dimensions may
be ascribed to the fact that the plane of polish cuts the pores at different
latitudes.
*
-
-129a-
Figure 27. The top view shows a specimen heated in vaouo
to a porosity of 32.7%, polished and etched normally. The
bottom view shows the 9sme specimen after numerous etchings
and repolishings.
130
-
-
The specimen shown on the two photomicrographs of Figures (26)
and (27) was heated for 45 hours in vacuum.
whatever of spheroidisation.
The pores show no indication
Figure (28) shows a photomicrograph of a
specimen which was originally heated for 16 hours in argon at 8500C, and
then refired in vacuum for 24 hours at the same temperature.
The density
rose from its original value of 4.74 to 5.90 after the argon treatment,
and after the refiring dropped again to 5.29.
This is to be expected from
the calculations, since the argon, originally at 106 dynes per square centimeter in the pores, was compressed by pore shrinkage to 72% of its volume,
that is, to a pressure of 1.39 x 10 6 dynes per square centimeter.
On heat-
ing in vacuum, as is shown in Figure (15) the pores must expand again, if
the theory is correct, and the pores must become more nearly spherical.
Figure (28) does how a tendency towards spheroidisation.
The relatively
small extent of the expansion of the pores does not permit them to assume
a fully spherical shape.
Figure (28). Specimen Heated at 8500C in Argon for 16 Hours, Then
at 85000 in Vacuum for 24 Hours. Note tendency for pores to assume
spherical shape upon expansion. Magnification 1001; 50% NH4OH-50%
H20 2 etch.
-
- 131
Figure (27) shows a feature of interest in that there is evidence
in several grains of growth across the original particle boundary.
This
feature is brought out in greater detail in Figure (29), which is taken
at a magnification of 5001.
There is in addition some indication of the
deformation of the straight twin boundaries, but this is possibly a result
of polishing.
Figure (29). Specimen After Heating for 45 Hours in Vacuum at
850OC. 50%NH4H-50%H202 etch. Magnification 5001. Note grain
growing across original particle boundary.
6
-
- 132
V.
Discussion of Results
The problem of .sintering is narrowed down to that of the consoli-
dation or densification of unpressed metallic powders in order to permit
differentiation between those phenomena which are due to cold work or
thermal effects during compression and those phenomena which are intrinsic
to the heat treatment of powder compacts.
Powders of only one component
are chosen, so that complications arising from the presence of a liquid
phase or from the processes of diffusion of dissimilar metals are ruled
out.
Although metallic materials are studied in particular, many of the
results are applicable to the consolidation of non-metallies such as are
of interest to geologists and to students of soil mechanics.
Copper is
chosen as the metal around which the computations are centered, because
it is one of the best known metals physically and is easy to handle in the
pure state.
A discussion of transient effects such as recrystallization reveals that at temperatures usually employed in sintering all phenomena
due to any previous cold-working of the powder take place during the relatively short heating-up period before measureable densification occurs.
Furthermore, above 4500 C all volatile impurities and adsorbed or chemisorbed gases are expelled from the metal and are present in the pores as
gas.
Only increases in pressure due to contraction of the pores are cap-
able of reversing this expulsion by introducing equilibrium conditions such
that compounds are formed during the sintering period.
For this reason a
temperature of 850 0C is selected for the calculations and experiments on
the sintering process as distinguished from transient phenomena ascribable
to surface or volume reactions of the metal with surrounding gases.
133
-
-
A preliminary discussion of the thermodynamical aspects of the
sintering process reveals that finely divided metal possesses a surface
free energy of the magnitude of the free energy changes usually associated
with chemical reactions if the particles are of smaller diameter than
about a.10th of a micron.
Use of a measurement of electromotive force
should give a better indication of the affinity of the sintering process
considered as a thermochemical reaction than the usual particle size determination by mechanical means.
Using the Gibbsian concept of a surface
phase between the solid metal and its vapor, a study of the reaction involved in sintering leads to the conclusion that two processes are taking
place side by side, one a spheroidization of the pores between the metal
particles, the other a reduction in the volume of the pores accompanied
by a general shrinkage of the compact.
The reason for this conclusion is
that from a thermodynamical standpoint pores are in equilibrium in a metal
provided they are spherical.
Spheroidization may take place by means of
surface diffusion or by evaporation and condensation.
pores can only take place if
Shrinkage of the
the solid takes on some of the characteris-
ties of a fluid, aid changes its shape by some form of plastic or viscous
flow.
A survey of the literature on sintering shows that it
is the
accepted view of modern investigators that the force involved in the
mechanism of sintering is, first, a force of adhesion which Is of the same
nature as the normal cohesive forces within a solid metal lattice, and'
second, the surface tension.
The force of adhesion has not previously
been calculated, and the surface tension has only recently been calculated
for solid metals, the result not having heretofore been applied to the
sintering process.
134
-
-
There is considerable confusion in the literature con-
cerning the distinction between mechanisms operating by surface diffusion
and those which take place by flow.
There is also much confusion in re-
gard to the transient phenomena due to the presence of gases and of proliminary cold-work.
Most of this confusion is clarified by the studies of
( ViA ~A GZC)
Httig and his collaborators, but the confusion between mechanisms of surface diffusion and of flow has not heretofore been conclusively cleared up.
(P1 4)
The mechanism suggested by Pines whereby the flow is one of
lattice defects moving outward through the lattice of the porous metal
is critically reviewed and it is found that his solution of heterogeneous
sintering has not hitherto been observed.
His solution of homogeneous
sintering is probably correct but does not lend itself to computation for
the purposes of experimental verification.
Frenkel's (Fr8) viewpoint
whereby the flow takes place by self-diffusion, whether of atoms moving
from lattice site to lattice site, or of defects moving in the opposite
direction, is similar, yet has the advantage of lending itself to experimental check.
The first detailed calculation presented in this thesis is that
of the force of adhesion between two metallic surfaces.
This calculation
is guided by the suggestion of Wretblad and Wulff and by the work of Bradley (brig)
(Frv)
and Frenkel. The field of electronic charge between two surfaces closely
approaching one another is studied, and it is shown that the mechanical
term introduced by the pressure of the electron gas is negligible compared
to the electrostatic term.
This is therefore computed for copper, and it
is found that it accounts for the experimental facts as reported in the
literature, namely, the presence of a force of adhesion acting at distances
135
-
-
as great as several hundred Angstrms, reaching a value of over a hundred
thousand pounds per square inch when the two surfaces are thirty Angstrbms
apart.
This force is not very dependent on temperature but is affected
by the cleanliness of the surfaces.
The effect of this force on two small spheres in contact is
studied, and it is found that plastic flow at room temperature takes place
to a very limited extent and can account for only about 10% of the shrinkage observed in sintering of very fine particles at elevated temperatures.
The tensile strength of compacts compressed at room temperature is satisfactorily accounted for by the calculated adhesion.
The question of the nature of the surface tension is next taken
up, and it is shown that this force and the force of adhesion previously
calculated are of like origin, with this difference, that in the surface
phase of the metal the non-electrostatic term due to the anisotropic
nature of the electron cloud is more important than it is in the space
between surfaces.
Samoilovich (SaL)) has performed the calculation of
the surface tension.
His results indicate that the variation of the sur-
face tension with temperature is very slight and that the surface tension
is only slightly higher in the solid than it is in the liquid state.
The stress imposed by the existence of surface tension on the
metal surrounding a small pore is calculated, and it is shown that for
pores of radius less than 10-6 centimeters the stress is equivalent to
the presence of a negative pressure inside the pore of about 35000 pounds
per square inch.
Accordingly the criterion of plastic flow in metals at
room temperature is examined to see if such small pores must become filled
up instantaneously by the normal mechanism of plastic flow by slip.
It is
136
-
-
shown that this is indeed the case if the pores have a radius less than
about 1O-6 centimeters at room temperature, and less than 1C5 centimeters Are- elevated temperatures.
It is also shown, however, that this
mechanism of the shrinkage of pores does not account for the shrinkage of
a compact, since at a short distance from the pores the metal is elastically and not plastically deformed.
Other types of flow are examined, and it is concluded that any
type such as slip or secondary creep which involves a stress barrier
beneath which the stress is unable to produce flow cannot be used to account
for sintering.
adduced.
Similarly no reversible type like primary creep can be
The only type of flow remaining is a viscous flow, and its
existence is studied in detail.
It is called viscous because, as in the
viscous flow of fluids, it involves a strain rate proportional to the
stress, and therefore can be dealt with by the use of hydrodynamical
methods of analysis.
The rate of flow is calculated by Frenkel and by
Kanter for solids under these conditions, and their results are the subject of experimental verification.
It is found in the experiments per-
formed here that the flow is slower than that calculated by either author.
A calculation is carried out to see if the spheroidization of
pores by transfer of material through the gas phase or along the surface
can take place at a rate comparable with the transfer of material by flow
through the body of the metal.
It is found that the gas-phase or surface
diffusion mechanism is very much slower than the volume flow mechanism.
In order to obtain sintering curves for compacts under various
conditions it is necessary to know if the shape of the pores is of great
consequence.
Two calculations are performed to answer this question,
and in both cases it
137
-
-
is concluded that for a reasonably isometric pore of
irregular shape there can be substituted a sphere of equal volume as far
as the calculation of the rate of shrinkage is concerned.
Accordingly
extensive computations, which are reported in the appendices, were carried
out to determine the rates of shrinkage of aggregates of copper particles
having pores of various sizes and under various atmospheres.
It is found
that in all cases a parameter here given the symbol F can be used to represent both the time, and the temperature, of sintering.
If sintering is carried out in vacuum, and there is no gas entrapped in the pores, then pores of all sizes shrink at all temperatures.
But there is a considerable length of time, at. the beginning of the process, during which the corresponding increase in density of the compact is
relatively slight.
rapidly.
After that period the shrinkage takes place very
Larger pores do not begin to shrink rapidly until later than the
finer pores.
This dependence of the time of rapid sintering on pore size
makes it imperative, in order to verify the theory experimentally, that
a material be used which can be packed in such a way that the pores are
all of uniform size.
Accordingly a narrowly sieved range of spherical
atomized copper powder of radius 10-02 centimeters was used in the experiments, and the results indicate that the number of pores and their average
size as calculated from the geometry and apparent density of the powder
mass closely represent the actual facts.
Experiments designed to verify
the curves calculated for vacuum sintering indicate that it is extremely
difficult if
not entirely impossible to attain the condition of vacuum in-
side the pores.
The reason for this conclusion is that the gases expelled
from the metal during heating-up are unable to escape before the pores are
138
-
-
sealed off from the outside atmosphere.
Extremely high pumping speeds
were used in an effort to extract the gas as fast as it comes out of the
metal, but the results nevertheless indicate that pressures of gas of
several centimeters of mercury remained within the pores.
Further computations were carried out, however, to determine the
sintering rates of aggregates of these spherical copper particles when a
pressure of one atmosphere of argon is maintained throughout the heat.
The pressure of argon in the pores at the moment when they become sealed
off from the outside is then known to be one atmosphere.
Experiments
carried out under these conditions substantiate the conclusions.
The
course of sintering is initially the same as in the absence of gas, but
after the pores have contracted to a certain size the gas inside them has
increased in pressure to the point where the surface tension is unable to
cause further contraction to occur.
and reaches a stable density.
The compact then ceases to shrink,
Such is the course observed experimentally.
From the experimental result it
viscosity coefficient of the flow.
is possible to determine the
This was done, and the value found was,
at 850 0C, 2.16 x 1010 seconds per centimeter cubed.
This corresponds to
a heat of activation of the flow units of about 86500 calories per mol.
The heat of activation according to Frenkel's theory of self-diffusion
should be that of self-diffusion for copper, namely, about 60000 calories
per mol.
The slightly higher values of density observed than were expected
from the theory are satisfactorily explained on the basis of the fact that
the pores are not all of the same size, and those that are finer than the
average cause a more extensive densification.
The experiments done in vacuo
139
-
-
support this explanation by showing evidence of the existence of larger
pores than average.
These pores, in accordance with the theory, expand
after the finer pores have contracted, and the density falls slowly during
long heating times.
-
- 140
VI.
Conclusions
From the calculations and experiments performed in the course
of the study reported here several conclusions may be drawn.
Some of
these have been suggested in the literature, generally without any experimental backing, often as alternative possibilities.
Here for the first
time definite quantitative data is presented in support of a comprehensive explanation of the sintering process.
It is found that the force of adhesion acting to cause metallic
bodies to weld together at low temperatures as well as at high temperatures
is satisfactorily accounted for by the electrostatic attraction exerted by
the superficial electronic configuration of one body on that of the other.
Thermodynamic analysis reveals that two processes are in action
simultaneously during sintering, one causing the pores between particles
to become spheroidal, the other causing them to contract.
Further calcula-
tion and experiments indicate that the spheroidization, which is accomplished
by surface diffusicn and transfer of material through the gas phase, is
much slower than the shrinkage, at least for copper at elevated temperatures.
The flow responsible for the shrinkage of pores and consequently
of compacts of powdered metals is shown to be of the nature of a viscous
flow, and not that of a plastic flow by slip.
The force causing the flow
is shown to be the surface tension, which is made up of two terms, one of
electrostatic origin, the other of mechanical origin, both due to the anisotropic distribution of electrons in the surface phase of metals.
The elec-
trostatic term is the same one found responsible at a distance from the
surface for the adhesion of metal surfaces.
Calculation of curves representing the course of densification
-
- 141
of compacts of copper powder under various conditions show that the flow
is slower than that predicted by the Frenkel theory based on self-diffusion. (Frg)
The experiments are, however, in agkeement with the calculations if
the
heat of activation of the units of flow is about 85000 calories per mol
rather than the 60000 calories per mol required to activate self-diffusion.
The calculations and experiments show that in compacts which contain a range of pore sizes the finer pores shrink before the larger ones
do; that in compacts into which a gas has been sealed under pressure during a pressing operation the larger pores expand rather than shrink, while
the finer pores shrink as in the previous case; all pores containing gas
reach a stable dimension after a period of time, and after tht
period
further heating does not lead to further densification or expansion, unless the gas is able to diffuse out of the pores at an appreciable rate.
The theory satisfactorily explains why cylindrical compressed compacts
expand in the direction of their axis and simultaneously shrink in a radial
direction.
In general, this thesis points out .that sintering of an elemental
powder is a process of viscous flow under the influence of surface tension
and gas pressure, and obeys the laws of hydrodynamics;
the units of flow,
be they atoms or lattice defects or blocks of atoms, require a heat of
activation of some 85000 calories per mol. for copper.
As a consequence
in the ideal case where a complete absence of foreign gas exists complete
densification of pure metal powder compacts would occur below the melting
point of the compact.
The time required for such densification is primarily
temperature dependent, yet also depends on the initial pore size distribution.
The more practical cases of sintering phenomena described in the literature
are readily explained in an analogous fashion.
-
- 142
VII.
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Hu22
HUttig, G.
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Sorptionsmessungen; Zz. anorg. 252, 1943, 95.
Jol
Jones,
, Principles of Powder Metallurgy, with an account
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Ka5
Kauzmann, W._, Flow of Solid Metals from the Standpoint of the
it
Chemical Rate Theory; AIME 143, 1941, 57.
Ka8
Kanter, H., The Problem of the Temperature Coefficient of Tensile
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Kell
Kelley, K. K., Contributions to the Data on Theoretical Metallurgy.
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Kel4
Kelley, G. C., Metal Powders, Effect of Time Temperature and
Pressure on Density; Iron Age, 145, 1940, 36, Feb. 22.
K17
Kieffer, _., and W. Hotop, Pulvermetallurgie und Sinterwerkstoffe;
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Kn6
Knudsen, M. H. C., The Kinetic Theory of Gases; London Methuen
1934.
I,
Experiments with Rock Salt Crystals; C. R. URSS
Lu3
Lukirsk,_P
Ma24
Mathews, A. P., Relation of Molecule Cohesion to Surface Tension
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Ma25
Macaulay, J. M., On the Seizure of Surfaces, J. Roy. Tech. Coll.
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Me7
Meyer, Q.., and Eilender, W., Die Sinterung von Hartmetallegierungen;
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Me2O
Metals Disintegrating Co., The Field of Powder Metallurgy and
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Mi27
Millard, E. B., Physical Chemistry for Colleges; McGraw, New York,
1941, 5th ed.
Na3
Nadai,
_, and Wahl.A,_ L., Plasticity, a Mechanics of the Plastic
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Ofl
Offerman, E
Ph2
Phillips, H. B, Vector Analysis; Wiley, New York, 1933.
Pi4
Pines, B. Ya, Mechanism of Sintering; Jour. Tech. Phys. 16, 1946,
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Pr5
Price, G. H.a S., Smithells, 0.t J., and Williams, S., V., Sintered
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Ra6
Rayleigh, Lord, A Study of Glass Surfaces in Optical Contact;
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Rhines, F. _., Seminar on Sintering; AIME Chicago meeting Feb. 26,
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Ri2
Richtmeyer, F. K., and E. H. Kennard, Introduction to Modern Physics,
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Sa8
Sakmann, B
_, Burwell, J. T., Irvine, J
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Measurement of the
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Sa9
Samoylovich, A
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Sa19
Sauerwald, F., Einige Neue Versuche zur Herstellung synthetischer
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Sa20
Sauerwald, F., Die Herstellung synthetischer Metallkoerper durch
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Sa2l
Die ohne vorhergehende Kaltbearbeitung eintretende
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Sa22
Sauerwald, F., and Holub, T. E., Kristallizationen zwischen m'glichst
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147
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Sa23
Sauerwald, F., and S. T Kubik, Uber synthetische Metallkorper VI;
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Sa24
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Sa25
Sauerwald, F., and Elsner, G.,
Sa26
Sauerwald, F., and Jaenischen, E., Nber Festigkeit und Dichte
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Sa27
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ber die wesentliche Faktoren bei Kaltbearbeitung
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Sa28
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Schlecht, L
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-
-
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-
- 150
VIII.
Biographical Sketch
The author, Amos Johnson Shaler, was born in London, England,
on July 8, 1917, and is a citizen of the United States.
His parents are
Mary Johnson Shaler and the late Millard King Shaler, both American
citizens.
His primary schooling was in Belgium and in Switzerland.
Secon-
dary schooling was received at Ecole Centrale des Arts et Metiers, and
American School, both in Brussels, Belgium, and at the Hotchkiss School,
Lakeville, Connecticut, U. S. A.
He was graduated from the Massachusetts Institute of Technology
with the degree of S.B. in Physics in June 1940.
During a year's inter-
ruption after the first two years at the Institute he did research at the
Royal Observatory in Brussels.
As Secretary to the Singer-Polignac International Congress in
Astrophysics, he published two volumes of transactions under the title
&
Novae and White Dwarfs (Les Novae et les Naines Blanches), Paris, Hermann
Cie., 1941.
In 1942 he wrote two papers on the properties of diamond, pub-
lished in The Diamond News, Johannesburg, South Africa, under the titles
Some Electrical Properties of Diamond and The Combustion of Diamond.
In
August 1945 he read before the South African Society of Production Engineers
a paper entitled Powder Metallurgy and published in The Engineer and
Foundryman, Johannesburg.
In 1946 he prepared a translation of Bravaist s
classic paper on crystallography.
It is now in publication.
In March 1947
he participated in the Seminar on Pressing, American Institute of Mining
and Metallurgical Engineers in New York.
He is a member of the American Institute of Mining and Metallurgical
-
151-
Engineers, the American Association for the Advancement of Science,
the Crystallographic Society, The Society of Sigma Xi, and the Societe
Belge d Astronomie.
APPENDIX I
Calculation of the Lowest Free Eneryv for a Solid Crystal at Room
Temperature
From Seitz, (Se7), the removal of one atom from an ideal lattice
requires energy h, equal to the atomic heat of sublimation.
To remove n
But this removal introduces an entropy term
such atoms takes nh ergs.
determined by the number of ways in which the n vacancies may be distributed
S
Remembering that N4,
n
S = k( in N!
This entropy is
k ln
NIO
-
among the N sites of the lattice.
1, and using Stirling' s approximation (Bul5)
ln n! - ln (N-n)!) =
-
= k( N in N
N - n ln n + n = (N-n) ln (N-n)
-
N - n) =
= k n ln N/n
(Seitz (Se7) reports this in typographical error as S = k n in n/N.)
The free energy is then minimum when
nh - kTn ln N/n
is least, i.e. when
d/dn (nh - kTn ln N/n) = 0
which gives
-h
n/N = e
The heat of sublimation of Cu is 81.2 kcal/mol at 298 0 K so
-81200 x 4,185 x 107
-h
2.3x298 x k
6.023 x 10 4 x 300 x 1.379 x 10'-L x 2.3
-
n/N
5.923 x 101
= log n/N
6 x 1C-6b
The difference in free energy between an ideal crystal of 1 mol and a
crystal of minimum free energy is then essentially zero at room temperature,
and even at 400&0K it would only amount to about 4 x 10-12 calories.
APPENDIX II
Calculation of the Free Energy of Spherical Copper Powder
Fricke (Frl7) gives the following figures for copper:
heat of sublimation
81.53 kcal at 2500; lattice constant 3.608 A; surface energy of (111)
.
plane, 2535 ergs/cm 2; surface energy of (100) plane, 2913 ergs/cm 2
Frenkel (Fr9) shows that the equilibrium shape of a crystal is a slightly
deformed sphere (see also experiments of Lukirski (Lu3)) and is made up
of stepped surfaces of lowest surface energy.
Therefore, in the calcula-
tion below the value of 2535 ergs/cm 2 is used, and the area is taken as
that of the sphere, although this is rather an underestimate.
A sphere
of copper of radius r then has the surface energy
(41Yr 2 x 2535)/(4.185 x 107) = 7.612 x 10~ 4 r2 cal.
It also has a weight of (4/3
' r3 x density).
The density is calculated
from the atomic weight, 63.57, and the lattice constant as given above,
and works out at 8.989 gr/cm3 . So the weight is
8.989 X 1.333 Ar 3 = 37.65 r 3
The surface energy per mol is then, in calories,
(63.57 x 7.612 x 10-4)/(3.765 x
1 0 1)
1 = 1.285 x 10-3
r
r
The surface energy for an ideal sphere of 63.57 gr. is
4W(2 x 63.57)2/3 x 2535
(4 W x 8.989)2/3 x 4.185 x 107
1.079 x 10-3 cal.
For single atoms (gas) the surface energy per mol on this basis is
4 (3 x 63.57)2/3 x 2Q5 x 6.023 x 1023
(6.023 x 1023 x 41Y x 8.989)2/3 x 4.185 x 107
r cm
100
1r
100
10-1
101
10-2
10-3
10-4
10-5
102
1o3
io4
105
1.285 x 10~J i/r
0.001285
0.01285
0.1285
1.285
12.85
128.5
etc.
=
F K cal)
0.000206
0.01177
0.1274
1.284
12.85
128.5
91145 cal
APPENDIX III
CALCULATION OF THE FORCE OF ADHESION
First we compute B and -4Be2(kr)3/2 ; A
W
T
r x 108
kT
1
ev
300
30
4.l43xLO~ 1 4
1.6OJx1C-1 2
60
"
"
-4e2 (kT)3/
Wi ergs
2
5.4041x10 40
-.7,78x310
5.7966x104
"
6.0292x104 0
120
240
1000
5
ev
300
6.1601x106 0
"
1.381x10- 1 3
30
-4.73x10- 3 8
A
t
30
4.l43x10~1 4
60
""
120
"
8.005x10
12
-7.78x10-
39
"
30
1.381x1&- 1 3
60
"
4r
___
5.5861x10 39
"
5.6107x10 39
-4.73x10
"
"
38
5.4433x10 39
5.5233x10 3 9
__/Wg
12
42.8
2.361xi0
2
40.9
2.201xlC 8
1.648x10-12
39.8
2.116x10
9.74410- 6
1.625x101
39.2
2.071x10 1 8
-2535.642
1.344x10 6
1.773x10
12
12.82
2.361x101 8
-2780.433
2.544x10
6
.692x1,0-2
12.25
2.201x10- 1 8
-420.545
-451.090
1.44x10
7
5.4433x10 39
5.5233x1039
"
e2 /W
5,3542x10 4 0
5.7444x1040
60
240
1000
B(see below)
"
-469.191
-479.377
1.344x10- 6
1.773x10
2.544x10- 6
1.692x10~
4.944X10
"
6
1
18
18
-42.360
2.88x10" 8
1.229x10-
6
8.193x10
12
197.8
23.45x10-1 8
-42.982
"
2.429x10
6
8.100x10
12
195.5
23.11x10
18
4.829x10
6
8.053x10- 1 2
194.6
22.85x10
18
"
9629x10
6
8.029x10-1 2
193.7
22.75x10 1 8
"t
1.239x-6
2
59.3
23.45x10-18
12
58.6
23.1c10-1 8
-43.471
-43.662
-257.784
-261.572
"
2.429x10-6
8.193x10
8.100x10~
2
Appendix III
B~~
~
5.4041x10 40
l.49x10-6
5.568x1C6
5.7966x104
2.69x10-6
6.0392x104 0
5.09x10
6.1601x1040
9.89x10-6
*5.354+2x14o
1.49x10-6
6
5.433x09 1.2 6x10-6
9
2e
2___
-11.0811
(___
_______
-16.6217 e5.5406 27.702?
-10.8245 -16.2368
"
1.6704x0- 6
*5.7444x10 40 2.69x10-6
5.5233xj1X
O
"
45.4123
27.0613
-10.5476
-15.8214 45.2738
26.3690
-10.2591
-15.3886
25.6477
-11.6040
-17.4060 &5.8020
45.1295
29.0100
-11.3474 -17.0226 45.6742 28.3710
5.568x10- 6 -11.1539 -16.7309 45.5770
27.8848
2.46x106
-10.8534
-16.2801
45.4267
27.1335
4.86x106
-10.5677
-15.8515
+5.2838
26.4192
5.6107x1(39 9.66x10-6
-10.2693
-15.4040
+5.1347
25.6733
5.5861x1&39
5.4433x109 1.26x10
5.5233x1039
* Second term
1.6704x10
2.46x10
2 /8n
/
6
"
6
-11.6768 -17.5152 +5.8384 29.1920
-11.3863
-17.0794
28.4658
+5.6932
0.01
0' )+Di
Z
e
49.8650
2.39x10-1 7
3.47x10 5
5.04x,027
7.33x1049
-1.677x10-1 5
48.7103
5.80x10 17 2.58x10 5
1.15x1027
5.13x10 48
-4.369x10-15
47.4642
1.51xlO-1 6
1.88x10 5
2.34x102 6
2.91x10 4 7
-1.183x10-14
46.1659
4.09x10-1 6
1.35x10 5
4.4Ax10 2 5
1.46x104 6
-3.272x10~4
52.2180
4.94x10- 1 8
6.34x10 5
1.02x10 2 9
1.65x10 52
-2.188x10- 1 5
51.0678
9.49x10'
18
4.72x10 5
2.35x10 2 8
1,17x]0 51
-4.508x10
50.1926 1.86x10- 1 7
3.78x10 5
7.67x10 2 7
1.56x10 5 0
-1.314x10-1 6
5.25x10-17 2.67x10 5
1.36x,10 2 7
6.92x10 4 8
-3.762x10-1 6
47.5546 1.4xo1 6
1.92x10 5 2.63x102 6
3.59x10 4 7
-1.020x10-15
46.2119
3.94x10-n 1 6
1.36x105 4.71x10 2 5
1.63x104 6
-2.876x10- 1 5
52.5456
3.05x10-18
6.89x10 5
1.56x10 2 9
3.51x1052
-1.372x10-1
4.93x10 5 2.92x10 2 8
1.73x10 51
-3.804x1f-16
48.8403
1.2384 8.33x10-1 8
15
6
Appendix III
3
-D
C
-D4
4D/C4
C8
3.766x10 2 9
4.987x10 52
4.93x10- 24
2.396x10 7 9.226x102 8
3.747x10 51
5.24x10-23 2.74x10-45
-8.338x10
1.816x10 7
1.952x10 2 8 2.210x10 50
6.71x10- 22 4.50x1043
-1.763x10 7
1.332x107
3.783x102 7
1.132x10 49
9.57x10- 21 9.15x104
-3.4l9x10 6
3.463x10
4.812x]031
7.087x10 5 5
4.93x10-2 4
2.43x10~ 47
-4.438x10 8
2.764x10 8
1.189x10 3 1
5.389x10 54
5.24x10- 2 3
2.74x10
45
-8.603x10 7
3.294x106
5.772x1028
l.o69xlo52
2.52x10-24
6.36x10
48
-5.214x10 7
2.359x106
1.038x1028
4.809x1050 3.66x10-2 3 1.34W10- 45
-1.028x10 7
1.712x106
2.026x1027 2.518x10 49
5.58x10- 2 2
3.11x10
-1.828x10 6
8
8.71x10-21
7.59x10-41
-3.302x10
7.476x10 30 1.53Jx10 55 2.52x10 24
6.36x10-4 8
-5.444x10 7
3.66x10- 23
1.34x10- 45
-1.039x107
3.002x10 7
1.224x10 6
3.823x107
66W26
1.154x10
2.776x10 7 1.420x10 30 7.660x10 53
-3.402x10 8
2.43x10~ 4 7
43
4.ma
7
5
-D3 C4/192
-D4 C8
CWf1 /e2
-6.004x107
9.669x103
1.212x106
10.3402
9.3402
0.10706
-4.792x107
2.518x104
1.027x107
18.6679
17.6679
0.05660
-3.632x10 7
6.822x10 4 9.945x10 7
35.3233
34.3233
0.02913
-2.664x107
1.886x105 1.036x10 9
68.6340
67.6340 0.01479 7.7234x10 1 8
-6.926x10 8
1.236x106 1.722x109
10.3402
8
3.245x10 6
1.477x101
18.6679
17.6679
0.05660 1.9212x1Cf 18
-6.588x106
7.575xJQ2
6.799x104 43.7204
42.7204
0.02341
---
-4.718x106
1.979x103
6.444x10 5 85.3588
84.3588
0.01185
---
-3.424x106
5.888x10 3
7.831x10 6 168.6357
167.6357
0.00597
---
-2.448x106
1.661x10 4
8.758x10 7 335.1894
334.1894
0.00299 3.9385x10i7
-7.646x10 7
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7
Appendix III
15u, 3
15U23
I.
.
75u22
U
(5+17u+..)A5&l7 u. .)
I
+)2
Y
S
0.8625
12222.4920
0.0180
18929.
7.7005
47,.2663
0.1794
0.2400
82726.7685
0.0030
106443.
6.2052
65.0903
0.0670
0.0600
606538.4925
0
695484.
5.5552
89.8287
0.0250
0.0150
4640731.9350
0
4984964. 5,.2664
124.9531
0.0090
0.8625
12222.4920 0.0180
18929.
7.7005
47.2663
0.1794
0.2400
82726.7685
0.0030
106443.
6.2052
65.0903
0.0670
0.0375
1169491.8255
0
1307101.
5.4355
99.8938
0.0179
0.0075
9004973.4990
0
9540143.
5.2090
138.4084
0.0063
0.0000
70662792.4350
0
72773276.
5.1015
195.6510
0.0026
0.0000
559846891.2000
0
568228769.
5.0508
275.0114
0.0010
0.0375
1169491.8255-
0
1307101.
5.4355
99.8938
0.0179
0.0075.
9004973.4990
0
9540143.
5.2090
38.4084
0.0063
74.2918 0.3444 0.072x10 7 0.000x107
4.9485x10-3
0.000x107
7.8730x10-3
66.8535 0.0240 156.6822 0.0490 1.069x10 7 0.OOOx107
12.241J2x10-3
4.273x107 0.000x107
18.6053x10-3
0.043x107
4.9485x10-3
7
42.9915 0.0660 108.0818 0.1330 35.073x10 7 0.043x10
7.8720x10-3
27.0255
0.1650
42.9915 0.0660
101.6100
108.0818 0.1330 0.272x107
0.0090 226.5631 0.0180
27.0255 0.1650
74.2918 0.3444 9.182x107
176.6518 0.0359
0.013x107
O.000x107 14.0548x10-3
115.0275 0.0060 253.4359 0.0123
0.050x107
0.0 00x107 21.0622x10-3
76.7580 0.0180
0.000x10 7
31.4594X10-3
526.4864 0.0025
0.874x10 7 0.000x107
46.0464xlO-3
76.7580
0.0180 176.6518 0.0359
1.733x10 7 0.000x10 7
14.0548x1-
15.0275
0.0060 253.4359 0.0123
6.861x10 7 0.000x10 7 21.0622x10-3
171.8100 0.0030
251.4750
0.0015
* Third integral
367.4610 0.0056 0.216x107
3
8
Appendix III
12
(L--s/)
0.0302x10
r4(
, dtI
u
213/2
15/2
3
0.0120x10-3
0.0016x10-
-
0.0044x10-3
3
0.18097
0.00194
0.0053xl0-3
0.0054056
0.0001615
.
0.0302x10-3
0.17903
0.0120x1o- 3 0.32741 0.31641 0.01100 0.0302x10-3 0.035098
-
0.0033x10- 3
0.003762
---
---
0.0011x10-3
---
---
0.00044
0.000026
--
0.0005x10-3
0.0003x10 3
0.07629
0.07614 0.00015
0.0004x10- 3
-
0.0033x10-3
o.oo1x1r-3 0.14007 0.13916 0.0009
29u1
29u2
29u21
270.8658
3.1047
---
93u,
0.0025x10 3 0.0027482 0.0000539
93u2
932
868.6386 9.9566
---
1
512.3691 1.6414
1643.1147
5.2638
---
0.8448
3192.0669
2.7091
---
6289.9620
1.3755
2.7779
995.3757
1961.3860 0.4289
270.8658
3.1047
0.8662
---
868.6386 9.9566
---
1643.1147
5.2638 9.9696
1238.8916 0.6789
3972.9972
2.1771
---
2446.4052. 0.3437
7845.3684 1.1021
---
15590.201
0.5552
---
512.3691 1.6414
3.1088
4861.4353
0.1731
9691.4926
0.0867
0.1688
31079.6142
0.2781
0.54131
1238.8916
0.6789
---
3972.9972
2.1771
---
2446.4052
0.3437
0.5690
7845.3684
1.1021
1.8247
U12
---
--0.0008922
--0.011492
0.0000339
--0.0003849
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5
6
III rlpueddy
Appendix III
fixdi
-I
*.LUos
4024.9537
2.2861x1(- 6
2.1332x10-5
---
5.5984x,10-7
0.9888x10-5
---
1.1825x10~ 7
4.U99x10-6
1.8796
2.2949x10-8
1.5564x10- 6
4.2808
21741.0433 2.6900
139975.4o77
3.9227
1.8568
998724.1430 1.4319
4024.9537
4.2808
---
2.2861x10- 6
2.1332x10-5
2.8169
21741.0433
2.6900
4.2849
5.598410-7
0.9888x10-5
---
262513.7460
1.6864
---
1910188.2120
1.3452
14558946.8410
1.1731
113654309.0600
1.5468
22.8873
'Y2~
I
Q-L
6.9968x10
8
3.0212x1(-6
1.3282x1098
1.0986x10
6
---
2.4614c10~ 9
4.5776X,10
7
1.0867
1.1693
4.4310x10~ 1 0
1.8310x10~
7
262513.7460
1.6864
---
6.9968x1r-8
3.0212x1(-6
1910188.2120
1.3452
1.5748
8
1.0986x107 6
2cL terf
c-+-j4 (4A~
tnt,
(ZeL
C4th~d reAJ (4.
9.2014x10-3
1.2172x10
2
r-
Itervf*.ikraj )z
9.1318x10-
1.3282x10~
A4- tLU e8-41
5
---
2.6599x10- 5
4.5386x10-6
6.0445x10-6
1.6552x10- 2
7.6498x10
4.3995x10- 6
2.2920x10- 2
2.2286x10- 6
---
9.2014x10-3
9.1318x10-
5
2.1382x10-5
1.2172x10- 2
2.6599x10-
5
1.8368x10- 2
5.0950x1-
6
7.3157x10-n6
1,4778x,0- 6
2.9412x10-7
2.5371x10---
3.9716x10-7
2.3280x10- 6
2
6
3.5835x10- 2
5.3700x10
5.036Ox10-2
1.9897x10- 7
1.8368x10- 2
5.0950x10-
2
1.4778x10-
25371x0I-
7
1.8164x1c8.2693x10- 6
6
4.8987X,0-7
4.5386x10-6
9.1620x10- 5
6.0445x10-6
---
5.8671x10-8
4.6440x10- 7
1 .0810x10-9
6
---
7.3157x10-6
6
3.6661x10-n 6
29412x10-7
Appendix III
11
(fist
15rerw^
5.9051x10-5
5
oxlcrm 6
2.h4cLO'6
_o.oO21x1Om- 3
3
tO .0015x10- 3
41+3130x10-3
-O.0009x10- 3
-/4.3152x,0- 3
5.9051x10-5
5
.5789x10
2.3x,0-6
O-6
OAO-
Ox1,76
2.3x106
oxrao-6
2,,3xQ 6
0,0005x10 7
oxlo07
0.0044x.10 7
Wx07
00429x10 7
OxiO 7
Q.4471x10 7
O.0001x10 7
O.7331x10 7
o0.3600x10 7
0.0004x,0 7
0 .0022x10 7
o1ooooxi7 010 7
5
-O.0021x10 3
3
+O.O0l5x19' 3
-4.3060x10
--
--O.OO56xicr 3
-4.3205x10"' 3
+O.ooo5x1Q-3
-4+.3756x10 3
-o.o00oooc3
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-oooooxao 3
-4. 32O5xlOC3
to~ooo5x1o-3
-4309J1x..O3
-0.00x- 3
i0.0OOx0 3
6295.088x10 7
--
5713 .3,8x,07
-mm
o~OOOIXlo
7
---
-..
000x10-3
10.751x,0 7
13 .713x10 7
5043.422x1&'
16.181x10 7
7
J.8,o44x1o 7
-554+7.35Qx1Q
8212 .,LlxO7
124.,024x10 7
5894.9900x,07
158 .195x,0 7
28056.678x10 7
3.370x10 7
28356.635x,07
368x7
0
274664.089x10 7
3.766310 7
4.104x10 7
0 .0083x-107
---
o7
(sUwOi 3tvrrs
-O.OOOEOC1O7 3 -o.000o6,cjr3
-4.2574x10 3
0
oxio 7
0,03,7
O.0034x1 7 a
-4+.2574x10'- 3
4.3O6Qxloi
2.3x10-6
1.6&9x1Qo-
tsw af 31ttvW'
-
1*6069clo-
.S-Kw 3 ttrws
7
OX107
OX107
27036.585xlo 7
o.o/21Ao7
OX107
-
2929/4.315x:10 7
39 .U~x1O
oxlo 7
2866o.063xco 7
/.2 193x107
O.0378x..
oxJ..oa7
*Fourth
integral
t,7
7
Appendix III
12
Appendixl
-(4)2+(4)-l-(3)2+(3)1
IIz1
-(4+)?+(4)?'
'M
get
0.072x10 7
0.0005x10 7
---
0.2963
4.bc10-6
0.272x10 7
0.0044x10 7
---
1.4090
9x10-7
6.7328
lX10
0.0429X10 7
1.069x10 7
4.273x10 7
0.4470x10 7
9.139X10 7
0.7327x10 7
35.030x107
6.3622x10 7
0.013x10 7
0.000ox10 7
7
1x10-7
ox1o-7
0.2963
41x107
j.4090
9x10~7
39X10-7
0.0000x10 7
13.3134
1x107
---
0.050x10 7
0.0003x10 7
72.4653
0
0.216x10 7
0.0034x10 7
406.7834
0
0.874x10 7
0.0378x107
2250.0830
0
1.733x10 7
0.042110 7
6.861,x17
0.442,xw0 7
0.0105x107
0.0000x10 7
--0.0000x10 7
13.3134
72.4653
18x10~7
6.306x10 1 0
9.5531x10-l1
6x10~ 7
5.727x10 1 0
2.7778x10-1 2
1x10
8.5268x101/
Oxlo- 7
2.2672x10~ 9
18x10- 7
2.2672x107 9
9.5531X10-
7
Jx110- 7
5.661x10 1 0
5.570x101 0
8.346x101 0
17ixi~ 7
11
9.9726x10- 1 3
-
6.094x10 1 0
2.806x10 1 1
1x10-7
2.5712x10-1 4
0
2.836x10
:.2196x10-1 6
0
2,747x101 1
1.412x-17
0
).9726x10- 1 3
. 57 1 2 x 1 0-4
1x10- 7
0
ox10- 7
--1l10~7
2. 7 04xl0O
2.834x101 1
2.771x10-O
-
41.9860
---
ox10o 7
1x10~7
0
1x10
7
Appendix III
T
r(A)
300
/kT
Z0_.
9
30
420.545
1.4882xl0-6
60
451.090
2.8882x10-6
120
469.191
5.0882x10
240
479.377
9.8882x1O.6
"t
0.8704246-6
10000
487.377
6
"t
0.6740524-5
30
2535.642
1.4882x107 6
60
2720433
2.6882x10-6
10000
2937.143
400.2883x10-C6
30
42.360
1.2576x10- 6
5.56842x10-6
0.4226372-6
60
42.982
2.4576x10-6
It
0.5681216-6
120
43.471
4.8576x10-6
1 1000
5
C -
-A
300
400.2882x10
6
5.56842x.5-
0.4591967-6
0.5875968-6
"
1
13
6
0.7261481-6
1.67053x10- 6
It
0.1971518-6
0.3261580-6
0.4126136-5
0.7160769-6
.
"
6
240
43.662
9.6576xil-
10000
43.816
400.0576ox10
30
257.784
1.2576x10-C 6
1.67053x10-6
0.1611984-6
60
261.572
2.457610-6
I
0.3066828-6
10000
266.649
400.0576x10-' 6
It
0.4124885-5
5 1000
0.8653006-6
6
0.6739273-5
2.87870x10-*6
3.47379x105
2.38556X,0L
17
1.67239x10-15
3.00382x10 7
-6
2.58466x10 5
5.79043x10 1 7
4.35334x10- 1 5
2.39731x10 7
5.32290x10-*6
1.87867x10 5
1.50815x10-1 6
1.17935x10"14
1.812/2x10 7
7.42035x10- 6
1.34764x10 5
4.08576x.0-16
3.26437x10~- 4
1.32834x107
4.72120x10- 5
2.11810x10 4
1.05234x10-1 3
8.54811x0i12
1.57673x10- 6
6.34223x105
3.91988xl108
1.65657x10-1 5
3.30665x108
2.11913x10- 6
4.7189bx10 5
9.51642x10- 18
4.31480x10-1 5
2.63960x10 8
2.58591x10- 5
3.86711,x10 4
1.72918x10~
3.86898x9
4
8.46475x1O'
2
2.1226xLo 6
2.33544x10 7
Appendix III
--D c
.
-D 2 coA
2.64629x10-"6
3.77887x105
1.85315x1017
1.30832x10-1 6
6
2.70319x10 5
5.06248x10- 17
3.62659x106 2 .38903x106
5.20087x107 6
1 .92275x10 5
1 .40679x10-1 6
1.01924xlio-15
7.33332x10 6
1.36363x1x 5
3.94368x10-1 6
2.86982x1c- 1 5 1.22422x106
4.71984i10-5
2.11871x104
1 .05J43x10-1 3
7.67824x,10-
2
1.908811105
1.44943x10- 6
6.89926x10 5
3.04505x10- 1 8
1.30828x10-1 6
3.65693x107
2.02620x10-6
4.93535x10 5
8.31854x10~1
3.6265Ox10-16
2.65440x107
2.58517x10-5
3.86821104
1.72768x10-14
7.67807x10-13
2.12084x106
D 1/c
-WI2/e2
3.69932x10
C8
C4
3.29136x106
1.71862x10 6
2
4.90507x10-24
2.40597x10- 4 7
3.40951x1098
6.00764x107
1.03277x101
5.22212X10-23
2.72705x10-45
8.33635x10 7
4.79462x10 7
1.86554x101
6.70281x10
4.49277x10-43
1.75949x107
3.62484x10 7
3.53108x10
9.13982x10"41
3.41453x106
2.65668x10 7
6.86216x101
2.56739X10-1/
6.59149x1028
3.32949x10 2
4.24522x106
277.78995x101
4.90507x10-24
2.40597x1O-47
3.37726x108
6.61330x1O8
1.03277x101
23
2.72705x10-45
8.26254X107
5.27920x108
1.86554x10
2.56739xlO-34
6.59149x10-28
3.29703x102
4.67088x1o7
277.78995x101
2.50133X10-24
6.25665x10i-48
5.23050x10 7
6.58272x10 6
4.36371x101
3.64792x10U
1.33073x10-45
9.94153x10 6
4.77806x106
8.52756x101
5.56784/x10-22
3.10008x10-43
1.83058x10 6
3.43724x106
16.85526x101
8.69915x10-21
7.56752x10-41
3.29897x105
2.44844x106
33.51066x10
2.56148X10l14
6.56118X10n28
2.99758x101
3.81762x105
1388.14958x10
2.50133xio-24
6.25665x10-48
5.23034x107
7.31386x107
4.36371x10
1.33073x1-45
9.94126x106
5.30880x1o7
8.52756x101
2.99751x101
4.24168x106
9.56024x10
5.22212x10-
3.64792x10-
21
23
2.56148x10-4
6.5611SX10-
28
1388.14958x1l1
15
Appendix III
15-11L4.1p/2di
III~
U,
(2).
U2
l/ul=u2
0.32743
1.25437
1.79973
1.0812x10 8
17.6554
5.66399x10-n20 0.23799
1.33715
2.86465
1.3735x108
34.3108
2.91453x10 20 0.17072
1.40170
4.45590
1.6162x10
9.3277
1.07208xiO1 1
67.6216 1.47882x10- 2
0.12161
1.44979
6.77341
1.7995x10 8
2776.8995 3.60113x10
4
0.01898
1.55182
51.14518
2.1712x10 8
1.07208x10
1
0.32743
1.25437
1.79973
11.9022x10 8
17.6554
5.66399x10- 2
0.23799
1.33715
2.86465
15.1231x10 8
2776.8995
3.60113x10~4
0.01898
1.55182
51.14518
23.8893x10 8
42.6371
2.34537x10- 2
0.15315
1.41884
5.11086
0.3364x108
84.2756
1.18658x107 2
0.10893
1.46229
7.71791
0.3688x108
167.5526
5.96828x10" 3
0.07726
1.49369
11.45031
0.3936X10 8
3
0.05471
1.51614
16.76286
0.4104x0o
1.56231
116.25769
0.4438x10 8
1.41884
5.11086
3.7380x108
7.71791
4.0973x108
116.25769
4.9313x108
9.3277
334.1066 2.9930610
13880.4958 7.20435x10- 5 0.008488
2.34537x10-2
0.15315
84.2756 1.18658x10-2
0.10893
42.6371
1.46229
13880.4958 7.20435x10-5 0.008488 1.56231
U3/2
log u1
1/2
48 1/2
0.4848873
3.0541
2.61944
148.3690 1.2468776 0.6234388
4.2018
1.90392
42u, 3/2
56.9754 0.9697746
28.4877
74.1845
log u1
200.9789
401.9578 1.5354309
0.7677154
5.8576
1.36576
556.0659
1112.1318 1.8300854
0.9150427
8.2232
0.97288
146334.2730
292668.5460 3.4435602
1.7217801
28.4877
56.9754
74.1845
148.3690
146334.2730
292668.5460
* Second term of Pi
---
52.697
0.15184
---
3.0541
2.61944
---
4.2018
1.90392
52.697
0.15184
16
Ap-pendix III
log
p1/2
*8u 1/2
1.6297876
0.8148938
6.5297
1.22520
1547.3338
1.9257019
0.9628509
9.1802
0.87144
2168.8009
4337.6018
2.2241500
1.1120750
12.944
0.61808
6107.1345
12214.2690
2.5238855
1.2619428
18.279
0.43768
1635400.0152
3270800.0304
4.1424050
2.0712025
278.4075
556.8150
---
6.5297
1.22520
773.6669
1547.3338
---
9.1802
0.87144
3/2
+2u 3/2
278.4075
556.8150
773.6669
log u1
1635400.0152 3270800.0304
117.82
8u,/2
U5/2
24.4328
265.7247
106.2899
33.6144
1309.7570
523.9028
46.8606
6895.7468
2758.2987
65.7856
37602.0659
15040.8264
421.576
406355569.5266
0. 4u 5/2
162542227.8
24.4328
265.7247
106.2899
33.6144
1309.7570
523.9028
421.576
406355569.5266
162542227.$
52.2376
11870.4884
4748.1954
73.4416
65201.2422
26080.4969
103.552
363388.2297
145355.2919
146.232
2040433.9435
816173.5776
942.56
22700163039.5274
9080065216.
52.2376
11870.4884
4748.1954
73.4416
65201.2422
26080.4969
942.56
22700163039.5274
117.82
9080065216.
0.06790
0.06790
17
A-pendix III
,:UOOOOA)
1
300
185.0787
6.31028x10 1 0
6.321x101 0
8.78x10 9
127185
30
703.9823
5.86864x10 1 0
5.882x10 1 0
4.39x109
63593
60
5.64049x100
5.657x101 0
2.14x10 9
31000 120
5.53761x1010
5.556x1010
1.13x10 9
16369
240
5.42159x10 1 0
5.443x1010
185.0787
6.25059x101 0
6.370x1010
7.62x10 9
110382
30
703.9823
5.81668x101 0
5.968x10 1 0
3.60119
52149
60
5.36873xl1 0 0
5.608x1o 10
3205.752
16217.77
162835317.7
1000
162835317.7
5
2.80147x10 1
28.018x10 1 0
8.06x109
116756
30
2.75384xlO
27.542x10 10
3 .3010&
47803
60
149795.8
2.74213x10 1 1
27.425x1010
2.13x109
30855 120
828533.6
2.73331x1011
27.337x1010
1.25x109
18107
240
2.72280x10 1 1
27.212x101 0
2.80138x10
28.051x1010
7.74x109
11210
30
2.75377x1011
27.579x10 1 0
3.02x119
43747
6C
2.72274x1011
27 .277x10 10
5356.023
300
27700.40
9083336959.
1000
5356.023
27700.40
9083336959.
1
rx1 8
T
10000
300
10000
5
4r+e2 /w,
300
3(2-
i
400.14410-6
0.00577x10 12
1.607x10-12
38.8
2.037x108
6
0.00577x10- 1 2
1.607x10'
11.6
2.037x108
1000 400.144X10-
10000
C0
.
r/ze
400.029x10- 6 0.00577x10~
10000 1000 400.029x10
B
-A
.2629x1040
6.2020x10 4 0
5.6305X1039
487.377
2937.143
43.816
E.6305x1039
266.649
6
2
2
0 .00577 x]Qf1L
8.011,x10 12 193.4 22.67
8.01x10-1 2
58.0
22.67
APPENDIX IV
/2
tan'2/A
2
r!
2a sic,
2rl 5in2 o, R 2
F = 2
)
Calculation of Total Force F in Fige
S oc
BC c
2
A
B :Sin-l (A sin K
A
sin(v' + sin' 1(7 sin-<))
sin c<
1
l
Ar
2
,4
1l
Cos<
2oc
When A is small,
2
A -
sin2e
24/
10
5C = r(
A2 sin
-1-
O
A2
2
1 8' Cos C<
cosAO
t
.
-the error is less than 1% for A'
3.
So the total force is now
S4'7rd rl
s
_ta'
3
sinoc do( (1
A cosA
2
a
A
sinc(1 + A cosec )/ doc
f
2
tan-1
when c :tanl, x2 1
A
when oc :0
sine< d o: dx
x1A
R2
3
W u 2r
R2
ctn tan 1 la
A
A2 A
2 I~4-~V
7'
a
A
Let 1 + A cos K = x
2
-i
+ 81to ri
d ae
.
SR2
tan-2
1rJ/21
3
,x = 1 +A
2
fx4df
sinec (1 + A coset )4 d o(=-:
2
"2
5A (
5
A2
4 2p)
'4
-
(1
4.
)
12
+
2
A
2 fT+XA
21o
sin
R2
Appendix IV
2
When A is small, this is equal to
A ( } +1
4 R2
( A,
R2
2
V
Volume of sphere:
- I
Volume of cylindrical pore:
r
:
1
) .
8 'yr 1
R2
1
3
V
AIYr 3 +
)
F-
3
.
within 3% for A
Yr
4
\J2(r
~0
106
= 3.16228
10-2
10-3
1o-4
10-5
10-6
3.16x10 3
1x10 3
3.16x102
1x1o2
3.16x101
1T-2
1x10 2
-6
a-or 2-
4*17x1c- 6
4.17x10 7
4.17x10-8
4.17x10- 9
4.17x10-10
r fJ2E
4.899x10-1
4.899x10- 2
4.899x0-3
4.899x10-4
4.899x10-5
f ,d
I4r
pcr
109
=
APPENDIX V
CALCULATION OF SINTERING CURVES FOR THE CASE OF GAS ENTRAPEN7
10-5
10-4
10-3
10- 2
a or r
48.99
3.162 x101
jxQO
3.16x10- 2
111o- 3
3.16x1C- 5
1x10o
1.345x10-4
1.03147
6.45x10-1
6
1.10303
2.04x10- 2
4.253x10-
1.345x10-7
4.253x10-9
1.345x10-10
1.36664
2.92079
-4.64173
4.253x10-6
2.92079
2.4x10- 6
6.45x10~ 4
2.04x10-5
6.45x10- 7
2.04x10- 2
-1.691
4.253x10-12
-4.64173
-1.512176
-1.137994
-1.041675
2.4x10- 3
6.45x10- 4
2.04x10- 5
6.45x10-7
2.04x10-8
-3.191
-4.691
-6.191
-7.691
1.345x10 7
4.253x1X-9
1.345x10- 1 0
4.253x1X-1 2
1.345x10-13
-1.137994
-1.041675
-1.01298
-1.00409
-1.001307
2.4x10 0
6.45x10-4
2.04x10-5
6.45x10- 7
2.04x10-8
6.45x10- 10
-3.191
-4.691
-6.191
-7.691
-9.191
9
4.253x1r1.345x10- 1 0
-1.00409
-1.001307
2.4x10 3
2.04x10-5
6.45x10~ 7
-4.691
-6.191
-1.000408
2.04xlO- 8
-7.691
-1.000122
6.45x10-
10-3
10-4
10-5
10-6
3.16x10 1
1x10 1
3.16x10 0
1x100
3.16x10lxlc- 3
3.16x10- 5
lxl1- 6
1.345x10- 7
4.253x10-9
1.345x10- 1 0
10 3
10- 2
io-3
1o-4
10-5
10- 6
3.16x10 0
1x10 0
3.16x10- 1
ixiO-l
3.16x10- 2
3.16x10- 2
1x10- 3
3.16X10- 5
1xi0- 6
3.16x10- 8
100
1T2
o-3
1x10- 1
3.16x10-2
io-4
jx10-2
10-5
3.16x10- 3
3.16x10- 8
1.345x10- 1 3
1 x 10-3
lxcr-9
4.25x10-1 5
-1.000004
2.04x10 - 1 1
10
-w
.
IM
L i Ihli
2
1x1o' 3
3.16x12-5
1x10-
-0.191
-1.891
-3.191
-4.691
-5.191
4.25x10- 1 2
10
-9.191
-10.691
PO
Appendix V
QL
10-2
r
6.3x10-4 -2.986x101.6x10- 3 -7.738x10-
6**
7** I%.(a-r)
0.9410
0.9789
-0.061
-0.021
2.586x10- 1 0
9.072x10--1
0.9878
-0.012
5.223x10~-
0.9939
1.0000
-0.006 2.624x10- 1 1
0.000 0.000
5.239x10~
2.580x101.217x10-
6.3x1c- 3
-1.950x10'-3.076x10-1
1.969x10- 1
3.096x10- 1
3.2x10-5 -1.536x10- 3
6 .3x10- 5 -3.055x10- 3
1.25x10-4 -6.092x10- 3
1.599x10-3
3.118x1(- 3
6.155x10- 3
-0.9605
-0.9797
-0.9897
0.9618
0.9810
0.9910
-0.039
-0.019
-0.009
-0.9948
0.9962
-0.004
-0.9974
0.9987
-0.2211
0.2212
-1.222x10- 2
2
1.228x10- 2
2.453x10- 2
5x10 4
-2.446x10-
3.2x10-
-0.568x10- 6
2.568x10-
2x10-5
5x10-5
8x10-5
-9.78810-9
-2.448x10-3
-3.918x10-3
9ix0-5
-4.408x10- 3
9.808x1o- 4
2.450x10-3
3.920x10-3
4.410x10-3
10-5
4.C10-8
1.8x10- 7
5x10- 6
6.3x10- 6
8x1o-6
-1.928x10 6 1.991x10-6 -0.9682
-8.787x10- 6 8.850x10- 6 -0.9929
-2.449x10- 4 2.450x10- 4 -0.9997
-3.086x10- 4 3.087x10-4 -0.9998
-3.919x10-4 3.920x10- 4 -0.9998
10-6
3.2xf ~ 8
lxlO
-1.567x10-6
-4.898x10-6
-1.568x10-5
-3.086x10-5
-3.919x10- 5
10- 4
3.2x10- 7
6.3x10~
8x10- 7
*
7
*
6
1.569x10-6
4.900x10-6
1.568x10-5
3.086x10-5
3.919x10- 5
F log
F
5*
-1.215x10- 1
1.235x10-1
2
4
2.5x1c- 3
2.5x10-4
>&
(***** n** Second term in F
3.18 6 x10-2
7.938x0-2
dyne/CM2
-0.9372
-0.9748
-0.9838
-0.9898
-0.9935
4x10- 3
0-3
2
2
=1
3.9&x10- 6
3.50x10- 6
3.12x10-6
2.50x10- 6
1.54x10-6
4.03x10~ 7
3.90x10- 7
3.65x10- 7
-0.001
0.172x10-12
2.08x10- 7
-1.509
6.416x10 1 2
4.17x10~ 8
4.17x108
-7.38
6.932x10-15
1.744x10-15
4.253x10-16
1.701x10-1 6
3.33x10~ 8
2.08xlO- 8
8.33x10- 9
4,17xj0-9
3.33x10- 8
2.08x10- 8
8.33x10- 9
4.1710-9
-7.48
-7.68
-8.08
-8.38
4.15x10-9
4.09x]-0 9
2.08x10- 9
1.54x10- 9
8.33x10- 10
4.15x10- 9
4.09x10- 9
2.08x10- 9
1.54x1O 9
8.33x10-1 0
-8.38
-8.39
-8.68
-8.81
-9.08
0.518x10-1 2
3.12x10
7
0,9683
0.9930
0.9999
0.9999
1.0000
-0.032
-0.007
-0.000
-0.000
-0.000
4.332x10-1
9.482x10-1 5
1.883x10- 1 7
1.076x10- 1 7
5.380x10- 1 8
0.9987
0.9996
0.9999
0.9999
0.9999
-0.001
-0.000
-0.000
-0.000
-0.000
5.401x10-19 4.03x10- 10
1.744x10- 19 3.75x10-10
5.529x10-20 2.83x10- 1 0
2.552x10-20 1.5 4x10-1 0
2.126x10 2 0 8.33x10 11
*
-5.41
-5.46
-5.51
-5.60
-5.81
4.03x10- 7 -6.39
3.90x10- 7 -6.41
3.65x10- 7 -6.44
3.12x10- 7 -6.51
2.08x10-7 -6.68
-0.9979 0.9984 -0.002
-0.9992 0.9996 -0.000
-0.9995 0.9999 -0.000
-0.9996 1.0000 -0.000
-0.9987
-0.9996
-0.9999
-0.9999
-0.9999
2
2
2
3.90x10-6
3.50x10-6
3.12x10- 6
2.50x10- 6
1.54x10 6
4.03x10-1 0 -9.39
3.75x10-10 -9.43
2.83x10-10 -9.55
1.54x10-10 -9.81
-10.08
8.33x10
PA = 1 dyne/cm2( continued)
Appendix V
a
r
10-2
8x,0 3
32x1c- 5
2.5x10- 5
10- 3
8x1o- 4
4
-0.568x2.- 3
-0.225x10- 3
2.568x1-3
2.248x10- 3
5
6
3
-L(a-r)
7
8.33x1c- 7
4.15x10- 6
4.15x10 6
8.33x10- 7
415i0 6
4.15x10- 6
-6.08
-5.38
-5.38
---2.235
--3.006x10- 1 0
8.33x10- 8
4.16xio- 7
8.33x10- 8
4.16x10- 7
-7.08
-6.38
-0.2154 9.2154 -1.535
2.065x1c- 1 3
4.17x10- 9
4.17x10-9
-8.38
0.2220 -1.505
0.1014 -2.289
-0.1069
-.0.1070
8x1o-7
7.081x1O-
1o- 5
1x10-9
1.737x10 8
8.61x10-8
10-6
3.2x10- 11 0.568x10- 9
2.568x10
9
-0.2211
0.2211
-1.509
6.418x10-
LO-2
lx1o-3
4.999x1c-
2
-0.9600
0.9639
-0.047
2.006x10- 1 0
4.799x10- 2
log F
--6.401x10- 9
9.733x10-9
---0.2211
-0.1010
.-.
0.757x10- 5
5
F
15
4.17X10- 1 0 4.17x10-1 0 -9.38
3.75xj0- 6
3.75x10-
6
-5.43
P0
Appendix V
'~
LO-2
1x10-3
1.6x10-3
2.5x10- 3
4x1o- 3
6.3x10
1o- 3
3
LO-5
-1.737x10- 2
-4.676x10- 2
-9.085x10- 2
-1.643x10- 1
-2.770x10- 1
3.2x10- 5 -0.563x10- 3
6.3x10- 5 -2.086x10- 3
1.25x10-4 -5.124x10- 3
2.5x10- 4
LO4
45
)-
-1.125x10- 2
5x10-4
-2.350x10-
1x10-5
-1.737x10~4
2x10- 5
5x10- 5
8x10- 5
9x10- 5
4X10- 8
1.8x1C- 7
5x10-6
6.3xi- 6
8x10g
2
4
03 dynes/cm2
6
7
oF
-0.2154
-0.4251
-0.5896
-0.7221
-0.8141
0.2452
0.4838
0.6710
0.8217
0.9264
-1.406
-0.726
-0.399
-0.196
-0.076
1.891x10- 7
9.766x10-8
5.366x10- 8
2.641x10- 8
1.028x10- 8
3.750x10- 6
3.500x10- 6
3.125x10-6
2.500x10- 6
1.542x10- 6
3.94x,10-6 -5.40
3.40x10- 6 -5.47
3.07x10- 6 -5.51
2.47x10- 6 -5.61
1.53x10- 6 -5.81
3.257x10- 3 -0.2211
4.086xl0- 3 -0.5106
7.124x10- 3 -0.7192
0.2303
0.5319
0.7492
-1.468 6.245x10- 9
-0.631 2.685x10-9
-0.289 -1.228x10- 9
4.033x10- 7
3.904x10-7
3.646x10i- 7
3.97x10' 7
3.88x10- 7
3.63x10-7
-0.8490
0.8844
-0.123
0.522x10-9
3.125x10- 7
3.12x10- 7
8.061x10- 2
1.100x10-1
1.541x10-4
2.276x10-1
3.402x10-1
1.325x10- 2
2.550x10- 2
-0.9216
0.9600
8.06bx10- 4
-0.2154
-6.40
-6.41
-6.44
-6.51
-0.041
0.173x10- 9
2.083x10- 7
2.08x10~ 7
-6.68
3.750x10- 8
3.77x1(~ 8
-7.42
-9.482x10- 4 1.011x10- 3 -0.9375
-2.418x10- 3 2.481x10- 3 -0.9745
-3.888x10- 3 3.951x10- 3 -0.9840
-4.377x10- 3
4.44x10-3 -0.9858
0.2182
-1.522
2.047x10-1 0
0.9497
0.9871
0.9968
0.9986
-0.052
-0.013
-0.003
-0.001
-0.96x1Q- 6 2.960x10- 6 -0.3242
-7.818x10- 64 9.818x10- 46 -0.7963
-2.440x102.459x10- -0.9919
-3.076x10- 4 3.096x10- 4 -0.9935
-3.909x10-4 3.929x10- 4 -0.9949
0.3255
0.7996
0.9960
0.9976
0.9990
-1.123
-0.224
-0.004
-0.002
-0.001
6.94a1xo1-2
1.746x10- 1 2
0.431x10- 12
0.140x10-1 2
3.333x10- 8
2.083x10- 8
8.333x10- 9
4.167x10- 9
3.33x10- 8
2.08x10- 8
8.33x10- 9
4.17x10-9
4.775x10-12
0.951x10- 1 2
0.017x10-1 2
0.010x10- 1 2
0.004x1r-12
4.150x10- 9
4.092x10- 9
2.083x10-9
1.542x10-9
8,333xj0-1 0
4.15x10-9 -8.38
4+.09x10- 9 -8.39
2.08x10- 9 -8.68
1.54x10- 9 -8.81
8.33x10- 1 0 - 9.08
-7.48
-7.68
-8.08
-8.38
A~J.
~TV
-
.appencix V
p0 2
a
10-6
3.21-Q 8
lx10- 7
3.2x10 7
6.3x10-7
8xlo- 7
.,-
+n
I,)
45
-1.536xl0- 4 1.599x10- 6
-4.867x10-4 4.931x0-6
-1.564x10- 5 1.571x10- 5
-3.083x10 5 3.089x10- 5
-3.916x10- 5 3.922x10- 5
5
10' dynes/cm
67
(a-r)
F
log F
-0.9605
-0.9871
-0.9960
-0.99795
-0.99839
0.9618
0.9884
0.9973
0.99925
0.99969
-0.039
-0.012
-0.003
-0.001
-0.000
5.238x10-1 5
1.569x10- 1 5
0.364x10-15
0.101x10- 1 5
0.042x10-15
4.033x10-0 4 . 0 3 x10 -l0 -9.40
3.750x10L-10 3.75x10- 1 0 -9.43
2.833x10- 1 0 2.83x10-10 -9.55
1.542x10lO-10 1.54x10-10 -9.81
0.833x10- 10 0.83x10- 1 0 -10.08
0-2
8x10-3
8x10-4
-3.602x1c- 2
-0.757x10- 2
4.235xl0- 2
7.081x10-2
-0.8507
-0.1069
0.9681
0.1475
-0.032
-1.914
4.361x10-9
2.574x10- 7
0.833x10- 6
3.833x10-6
8.29x10- 7
3.58x10-6
-6.08
-5.45
0-3
8x10-4
25x10- 5
-3.819x10- 2
-o.225x10- 3
4.019x10- 2
2.225x10- 3
-0.9502
-0.1010
0.9898
0.1052
-0.010
-2.252
4.359x10-ll 0.833x10- 7
9.577x10- 9
4.062x10- 7
8.29x10~8
3.97xO- 7
-7.08
-6.40
5x1J-6
-2.133x10-4
2.766x10- 4
-0.7713
0.7813
-0.247
3.319x10'-3
3.959x10~ 3.96x10
8
-7.40
9.731x10- 12
4.156x10- 9
4.15x10-9
-8.38
10-4
jo- 5 25x10~ 8
-0.225x10-
6
2.225x10-
6
-0.1010
0.1014
-2.289
lx10- 9
-1.737x.10-
8
8.062x10
8
-0.2154
0.2157
-1.534 2.063x10- 1 3
jo-3 1x1o- 4
-3.899x1r-
3
5.899x10- 3
-0.6610
0.6885 -0.373
1.587x10~ 9
jo- 4
1x10- 6
-1.737x10-5
8.061x10- 5
-0.2154
0.2182
-1.522
2.047x10
10-5
1X10- 6
-4.799x10-
5
4.999x10- 5
-0.9600
0.9639
-0.047
2.006x10- 1 3 ,3.75Ox10-
]0-6
10
4.162x10-1 0
4.16x10-1 0 -9.38
3.750x10-7
3.77x10-7
4.125x1028
9
-6.42
4.15x10 -8
-7.38
3.75x10-9
-8.43
Appendix V
P
4
r
a
.-
4
10-5
-0.267x10- 3
-0.169x1c- 3
-0.1646
-0.1068
-0.0405
-0.0260
3.20x10- 5 -0.568x10- 3
-1.449x10- 3
5x10- 5
2.5x10-5 -0.225x10- 3
2.568x10- 3
3.499x10- 3
2.225x1X-3
4x10-5
-2.963x10-3
2.965x10-3
6.3x1O-
-2.086x1o-3
7X10-7
8x10-7
5x10-6
-0.267x10-5
-0.756x10-5
-133x10-4
-2.770x10- 4
-3.602x10- 4
6.3x10
6
6
3.2x10-
8
8x10-
10-6
10-2
10-
6
10-5
-1.247x10- 3
7.571x10- 3
7.081x10- 3
6.592x10- 3
6.494x10- 3
-'3 9x10"8x10-4
7x10- 4
6.8x10-4
11107
-4.757x10- 3
-0.568x,0-
7 -3.899x10-
6
6
-0.173x10~
4
7
(a-r)
F
log F
1.5 6 x10-7
3.82x10- 7
8.3 6X10- 7
1.03x10- 6
-6.80
-6.42
-6.08
-6.00
2.985x10-
0.833x10-
7.108x10- 6
8.989x10- 6
1.250x10- 7
1.333x10- 7
-1.096
0.6354 -0.453
0.1527 -1.879
0.4902 -0.713
0.7721 -0.249
1.474x10- 7
0.610x10- 7
2.528x10- 7
-6.75
-7.09
-6.55
1.21x10-7
-6.92
0.336x10-7
2.833x10~ 8
2.083x10- 8
3.125x10~ 8
2.503x10-8
1.541x10-8
1.76x10- 7
0.82x10- 7
2.84x10- 7
4.086x1o-3
-0.2210
-0.4202
-0.1010
-0.3242
-0.5106
0.49x10-7
-7.31
6.592x1lo-5
7.081xio-5
2.765x10-/4
3.403x10- 4
4.235x10-4
-0.0405 0.0461 -3.077
-o.1068 0.1215 -2.108
-0.7713 0.8777 -0.130
-0.8141 .0.9264 -0.076
-0.8507 0.9681 -0.032
1.309x10-8
8.965x10-9
0-.555X10-9
0.032x10- 9
0.013x10- 9
3.875x10-9
3.833x10-9
2.083x10-9
1.542x10-9
0.833x10- 9
1.70x10~8
1.28x10-8
2.64x10-9
1.57x10- 9
8.47x10- 1 0
-7.77
-7.89
-8.58
-8.80
-9.07
4.033x10- 1 0
2.568x10-
5.899x10-
6
6
1.979
1.784
1.612
6
6
0.417x10-7
0.020
2x10-2
1.6x10- 2 0.216
1.25x10- 2 0.387
1x10-
5
1.145x10- 6
1.668x104.086x10 6
1.080x10- 5
-0.22x10- 6
6
106 dynes/m
-0.269
-0.702
-1.671
-2.114
3.2x10- -1.468x10- 5
6.3x10 8 -2.086x10- 6
-8.798x10- 6
2x10- 7
2.5x10~ 8
=
2.225x10- 6
8.022x10- 5
0.7640
0.4957
0.1880
0.1208
0.3342
6
0.959x10-7
7
0.2302
-1.469
0.6884
-0.373
0.9167
0.5318
0.8487
-0.087
-0.631
-0.164
6.247x10- 1 2
1. 5 88x10L-12
0.370x10- 1 2
2.686x10- 2
0.698x10- 2
0.0102
0.1212
0.2404
0.0298
0.3539
0.7022
-3.513
-1.039
-0.354
1.494x10- 5
4.418x10- 6
1.504x10- 6
-0.1010
0.1052
-2.252
9.577x10~ 12
4.058x10- 10
1.891x10-10
3.750x1-T 9
-0.2210
-0.6609
-0.8800
-0.5105
-0.8147
-0.2154
0.2452
-1.406
3.750x10-1 0
2.833x10- 1 0
3.904x10- 1 0
3.333x10-10
-4.167x10-2.500x10-1.041x10-
6
6
6
4.10x10- 1 0
3.77x1-lO
2.84x10- 1 0
3.93x10-10
3.34x10- 1 0
-9.55
-9.40
-9.47
1.08x10- 5
1.67x10-6
4.62x10- 7
-4.97
-5.78
-6.34
-9.38
-9.42
4.15x10~ 1 0 -9.38
3.94x10- 9
-8.40
Appendix V.
P
a
r
0 .2
1.6 x10-1
1x10- 1
6.3x10 2
4X10-2
2x10-2
+
r
a
= 109 dynes/cm2
-
r
23.78
26.72
28.53
29.66
30.64
TlEr
39.46
36.52
34.71
33.58
32.60
4
0.1211
0.3423
0.5278
0.6722
0.7809
0.1336
0.3776
0.5822
0.7415
0.8614
-2.013
-0.974
-0.541
-0.299
-0.149
8.561x10 6
4.142x10-6
2.301x1O- 6
1.272x10-6
0.635x10 6
-6.25x10 6
-3.75xl0-6
-2.21x10-6
-1.25x10-6
-0.63x10-6
2.31x10- 6 -5.64
3.97x10 7 -6.40
9.23x10-8 -7.04
2.20x10- 8 -7.66
9.52x10- 9 -8.02
0.0121
0.0166 -4.098
5.512x10- 7
-2.21x10-7
3.30x10~ 7
0.1736 -1.751
0.3209 -1.137
0.4607. -0.775
2.355x10-7
1.529x10 7
1.042x10-7
-1.67x10-7
-1.25x10- 7
-0.92x10-7
0.442x10- 7
6.88x10-8 -7.16
2.79x10~8 -7.55
1.26x10-8 -7.90
-0.42x10-7
3.00x10-9
0.076x10-2
6.249x10- 2
0.713x10-2
1.203x10- 2
1.595x10-2
5.612x10-2 0.1270
5.122x10-2 0.2348
4.730x10-2 0.3371
2.182x10 2
4.142x10 2
log F
-6.25x10- 5 1.45x10- 6
-3.75x10- 5 3.71x10-7
-2.21x10-5 1.24x10 7
-1.25x10-5 3.7x1O-8
-1.42x1c-5 1.33x10- 8
5x10-4
4
4 x1o
2x104
F
6.395x.0- 5
3.787x10-5
2.221x10-5
1.254x10 5
0.418x10-5
6.3x10~ 4
3.2x104
(a-r)
-0.475
-0.282
-0.165
-0.093
-0.031
4
1o0-
7
0.6216
0.7546
0.8478
0.9110
0.9694
1.6x0-2
1x10-2
6.3x10-3
4x10-3
2.5x10 3
1.784
1.490
1.309
1.196
1.123
6
0.6026
0.7316
0.8219
0.8832
0.9398
10-3
0.216
0.510
o.691
0.804
0.877
5
0.5269
0.7201
-0.328
I-6
7.0x10-7 -0.267x10-5
8.0x10- 7 -0.757x10-5
9.0x10~ -1.247x10 5
10- 2
1.25x10-2
2.5x10-1
31.01
19.38
32.23
43.87
+0.9620
+0.4417
4xc10~
0.9923 -0.008 1.041x10-6
0.4556 -0.786 1.057x10-4
12.03
51.22
+0.2348
0.2422
-1.418
1.907x10-4
+0.8845
0.9756 -0.025
6.592x10-3 -0.0405
7.081x10-3 -0.1069
7.571x10 3 -0.1646
0.1880 -1.671 2.248x10- 10
0.4962 -0.701 0.943x10 10
0.7640 -0.269 0.362x10-10
-5.84
-6.43-6.91
-7.43
-7.88
-6.48
-8.52
1.25x10~ 10 3.50x10- 10 -9.46
0.83x10 0 1.78x10 10 975
047x010 0.83x0-10 -10.08
-1.041x10-6 0.Ox10-10
-4
6
-1.00x10- 5.73x10-5.24
-1.62x10-4
2.82x10-5
1.050x10~ 7
-1.04x10-7
8.83x10-10 -9.05
56.12
tO.1270
0.1310 -2.033 2.734x10~ 4
3.084
+0.9516
0.9815 -0.019 2.511x10-6
3.24
8
8
2
4 2.550x10 2 3.775x10 +0.6755 0.9232 -0.078 1.046x10 -1.04x10
1.25x10
to~
10~4 1.25x10~4 2.55OX10-2
0.9232 -0.078 1.046xlO-8
3.775x10-2 +0.6755
-2.04-x10~4
-2.50x10-6
6.92x10-5 -4.16
1.11x10-8 -7.96
8
1o-3
1.25x10- 3
0.939
10-2
5x10-1
7.128
6x10
2
1.061
-1.04%10-8
1 .llxlO
-4.55
-7.96
4.08x10~11-10.39
APPENDIX VI
FOR THE CASE OF COMPOUND FORMATION
VIOFE
AppendxIINT
P.
100
a
-el 5x102 5x10-2 10-13
5x1010-2
10
1o-3
10-4
o3
109
6
5x10-
10"
5xJ0
1Oo-3
5x10
5x10-4
1o-5
5x10-6
10-5
06
5xic-4
5x10- 5
10-5
10-6
i-1
P a
P r
r
5j10-7
---
-2400
5x10-3
5x10-4
5x10-5
5x10- 6
-5x10 2 2 .08x1 5
10-2
10- 3
10-4
-4
-----
-2400
---
---
-2400
-2400
----
--
-2400
---
-5x10-2
-5x10-5x10-4
-5x10-5
---
-2400
0
Sx
o2
-2350
-2300
1.002
0.009
55x10-2
2395
9
-239
10-1 -2400
-239
-23990
2
1.00
---
0.00
--- 1
2400
~
---
+97.6
0.707
0.797
0.133
---
09
0.0
0.001
--- -
-----
-7 105xl
5x--
5x054
11-
-2400 -2400
---2400
9 -8.68
2.08x1O7 2.08xlO
-10 -9.68
-5x10- -5x10-
-----
-5x10-6
-5x10-7
---
2.O8,4.68
2 .08x102.08x1o-7
-5.68
-6.68
2.08xio-9
2.08xi0-10
-8.68
-9.68
2.15x10-5
-4.67
2.08x10-8
-5x10- 4
5
2.08x10- 8
-5x10-
-7.68
.. ,68
-6.68
-7.68
--
-
105
104
103
+497.6
-47.6
-1900
- 7.6
-1400
5.098
6.263
1.357
-----
io-3
1o-4
10-5
1o"6
0-6
5x1
-77
5y0
5x10-
5
5
5x101l
102
101
-23
59
-23950
-239
00
-23
0
-239
-2399
2
1.02
1.002
101
5xJjyl
5x10 8
10 8
1x10 8
0.699
5x10-3
5x10
5x10
lxO0 7
5.000
10-2
10 7
5x10 87
5.001
0.699
-----
10-4
10-5
5x105
5x10-4
5xlo
5x10-5
106 x1- 510
105
104
13
497600
47600
-1900
9 600
7600
-1400
5.090
.098
0.700
0.707
---
1.357
0.133
--
7
log F
x10-74
2.09
2.o8xlO-6
5x0- -5x10
5x102
5x10 4
10-2
F
.
r-a
10-56
5x10-3
104
7
5x10
lo-3 0:2400
06 5x 5x1 5x1
5x10 21
-
fog-25.-zc
p?.-20-
-
-5x10- 7
-510-
--
1.63x10- 6
1.83x10- 6
-5.79
3.05x10 7
-6.52
2.19x10 9
2.0
10 1
5x10-8
-7.67
2 .09x
-8.68
8
1.61x0- 99
-8.79
1.61x10
-8.79
1.61x10- 9
1.63x10-9
-8.79
3.05x10"0
-..52'
.2.08xiO-10
-574
-9.6
-9.68
-8.79
2
Appendix VI
S
a
100
10-1
10-2
1o-3
1o- 4
03
10- 3
1o-4
io-5
03
iO-1
10-2
106
10 mi
r
.r-a
F
iog F
110- 2
-9x0-2
1x0-1
-1X10-1
-1x10- 1
-1x10- 3
-- x-o- 4
-110 5
4.17x10 6
3.75x10-5
4.16x0-5
4.17x10- 5
4.17x10- 5
4.17x10- 5
4.17x10-8
4.17x10 9
-5.38
4
4.17x.0-8
3.75x10 7
4.17x10- 7
4.17x1c- 9
4 . 1 7 x1)-1L
-7.38
-6.43
-6.38
-8.38
-9.38
----
4.34x10- 6
4.25x10-5
4.25x10-5
4.18x10m.7
-5.36
-4.37
-- 37
-6.38
9x10- 2
10-2
1o-4
1o-7
10_9
9x10-3
9x10-4
9x10- 5
9qx0- 2
10-2
10- 4
10-7
10- 9
9x10-3
4
9x1&9x10-5
10.1
10-1
10-1
10-1
10-1
10-2
10-3
10-4
-2400
-2400
-2400
-2400'
-2400
-2400
-2400
-2400
-2400
-2400
-2400
9x1c- 4
1o-4
10-7
9x10-1
10-1
-2399
-2399
9il0-5
9x1O- 6
9gx0- 2
-2400
-2400
-2400
-2400
-2399
-2399
9x10-3
1
1
1
10-1
10-2
9,0-2
10-4
10-7
9x10-3
90
10-1
10-4
9-
100
100
100
10
-2310
-2400
-24+00
-2391
-2300
-200
-2300
-2390
1.004
1.043
1.043
1.000
10-5
10-2
10
-2400
-2390
1.004 -0.002
---
4.17x10-6
-5.38
2x10-l
1.5x10-1
3.5x10~ 1
2x105
15x10 5
3.5xi0 5
1.1x10 5
10 5
105
105
+197600
+97600
97600
97600
2.025
1.512
3.561
0.306
0.179
0.552
---
147600
347600
---
7.05x10--7
4.14x10- 7
1.27x10-6
-6.15
-6.38
-5.90
105
107600
97600
1.102
0.042
---
0.97x10-7
-7.01
1.1x10-1
io-4
-2400
---
--
-2400
-2400
-2400
---
-2400
---
-
------
---
----
-9x10-4
---
---
-2400
-ix1O
---
-2400
---
---0.002
0.018
0.018
0.000
x1
3
-lx10-5
1.1y-----
---
6
-4.43
-4.38
-4.38
-4.38
-4.38
-7.38
-8.38
Appendix VI
106
3
a
r
10-2
1.1x10-2
1.5xQ 2
2x10- 2
3.5x10-2
10-1
9x10-4
10-5
10-7
9x10- 5
10-3
10- 4
10-6
1C7
10-5
9x10- 6
10-7
10-9
1o
6
9x10 7
10-8
109
10-2
10-3
10-4
10-5
10- 6
10~1
5x10-1
10-2
10-1
10-3
10-2
10-1
10-4
10-3
10-1
9x10- 7
10~
1or 7
P
Pn32r
(a
1o4
104
104
103
103
103
100
8600
12600
17600
32600
97600
-1500
-2390
-2400
-2310
100
100
-2399
-2400
7600
7600
7600
7600
7600
-1400
-1400
-1400
-2300
-2300
9
10
10-1
10-3
10
10
1
1
1.x104
1.5x104
2x104
3.5x10 4
105
900
10
10-1
90
1
10-1
9x10- 1
10-2
108
5x1o 87
10
108
106
107
108
105
10 6
1 8
9x10
10 4
1O4
107
107
106
10 6
105
105
10 5
104
iO4
10
103
r-a
-2
F
log F
1.24x10 7
5.05x10- 7
8.40x10 7
1.45x10- 6
2.55x10-6
6.90x10- 8
5.35x10-7
5.39x100 7
4.34x10- 9
-6.91
-6.30
-6.08
-5.84
-5.59
-7.16
-6.27
-6.27
-8.36
-7.38
-7.37
-9.38
-8.38
-8.38
1.132
1.658
2.316
4.289
12.84
1.071
1.707
1.714
1.004
0.053
0.219
0.365
0.632
1.109
0.030
0.232
0.234
0.002
---
0.018
0.018
---
-2300
1.043
1.043
---
4.21x10- 8
4.25x10 8
-2391
-2390
1.000
0.000
---
4.18x10-10
-2400
-2390
-2390
-2399
-2399
1.004
1.004
1.000
1.000
0.002
0.002
---
9997600 10.00
9997600
50.01
997600 10.02
997600 100.2
97600 10.22
97600 102.4
97600 102.4
7600 12.84
7600 131.3
7600 13357.
-1400
1.071
-2400
-2399.1
-2400
99997600
499997600
9997600
99997600
997600
9997600
99997600
97600
997600
99997600
-1500
---
-----
0.000
-----
0.000
---
1.000
1.699
1.001
2.001
1.009
2.010
3.010
1.109
2.118
4.119
0.030
---
---------
-------
-------
10
103
-2390
-1400
1.707
0.232
---
100
103
-2300
-1400
1.643
0.216
---
4.13x10-9
4.17x10-9
4.16x10- 11
4.12x10-10
2.30x10-9
3.91x10-9
2.30x10-9
4.61x109
2.32x10-9
4.63x10- 9
6.93x10-9
2.55x10 9
4.88x10-9
9.48x10-9
6.90x10 1 '
5.35x10'-0
4.96x10- 10
-10.38
-9.39
-8.64
-8.41
-8.64
-8.34
-8.63
-8.33
-8.16
-8.59
-8.31
-8.02
-10.16
-9.27
-9.30
APPENDIX VII
Calculation of the sintering curve of pores 10-2 cm. radius for inside
.
and outside pressures of argon of 106 dynes/cm2
-l r
r2dr_
2
2(r
(-PYa-'
3
-t r
P-
P,0
+
Po fa
)
F = -'
To integrate this expression we require the solution of the auxiliary
equation (for b, c, d):
(r 2 + br + c)(r + d)
+ r 3 + 2 r 2/pt ,
-P a3 /t
This leads to
b
4
22/P'
d
c + bd . 0
cd = -Poa3/pi
whence
b =-c/d
-
c/d + d = 2
d2 - 2
- 2
4 2
c.
0
d/P' - c
4r 2
2+P+ 40
Oc .,1
7
/P'
=-Pea
0
(
c
p3
Poa
c2 (4(2
4c)
4~ P
2
3
Pod
PI
0
0/
PO2a6
3c
C3 . 2 dPa
2
Pt
0
B
p1 2
-
0
P 2a6
4
0
A3
-
-6
3 P 3 a9
27Pb
4t
0
2
Appendix VII
Putting in the values of Po and P0 in order to determine the sign of the
discriminant and be able to proceed,
-B/2 = P2a6/2P 2 = 5 x 10-13
B2/A
=
3a/4
/
A 3 /27
2.5 x 10-25
3
O
-8
ax0
x10
.2
-8
18___-5,_2________
27 x 10
27P0
2.5 x 10-25 - 5.12 x 10-28 = 2.49488 x 10- 2 5
B2 /4+ A 3 /27
B2 /4 1 A 3 /27 = 4.99488 x 10-13
- B-
2
b
I
4
__
+ 427
5.12 x 10-16
9.99488 x 10-1 3
c
1.079984 x 10-4
d
-106/1.079984 x 1l~4
2400 x 10-6
2 f/P, - d
+
=-
9.2594 x 10-3
9.2594 x 10-3 = 1.16594 x 10-2
The integral now is
-176 r
F
F:-10
Ja
Let
2t169
02rd
(r2 i 1.6594 x 10-r '+ 1.08 x 10-4)(r-9.2594 x 10-3)
r2/(r 2 +br+c)(r+d) = A/(r.d) + (Br+C)/(-r 2 +br+c)
This leads to
A+ B
*1
Ac t Cd = 0
Ab + C + Bd
=0
Whence
-9,2594 x 10-3
9.29 x 10K.
-1.16594 x 10-2
-
9.2594 x 10-3
0.2842
Appendix VII
3
B
1 - 0.2842 = 0.7158
C
10-4
-0.28418 x 1.08 x
=
3
1ox
-9.2594
3.315 x 10-3
F
-10-6 [
rdr/(r4d) + Br
rdr/(r 2 +br+c) + C
A in(red) + B in(r2 +br+c) + (-B)
-10-6
2
r dlr/(r2+br4c
-
and
ta-1 2rab
r
.10-6 E. 2 84 18 in (r-9.2594x10- 3 ) + 0.71582 ln (r 2 +1.1659 10- 2 r+1.08x
L0
0.7406x10-3
3.2459 x 10-4
2
0.09978 (tan 1 2r + 1.1654 x 10
- ta-1 1.840)
1.7206 x 10-2
.
It is evident upon inspection that F becomes infinite at r = 9.2594x10-3
For values of r of 9.3 x 1i-
3,
9.4 x io- 3 , 9.6 x 10-3, 9.9 x i0-3, we can
set up the table of computations as follows:
9.3xi0
r
3
9.4x]0-3
9.6x10-
3
9.9xl1-
3
9.2594xi0-3
)
-0.82517
-0.47217
-0.22074
-0.04122
0.71582 in (
)
-0.04946
-0.04231
-0.02812
-0.00697
Fx1O 6
0.87272
0.51284
0.24779
0.04791
log F
-6.05914
-6.29002
-6.60590
-7.31957
-2.03342
log r
-2.03152
-2.02687
-2.01773
-2.00436
-2.03342
-
0.28418 in (
-0.05235