- Email: [email protected]
A unifying theory of the stability of penetrating liquid phases and sintering pores
A UNIFYING
THEORY OF THE PHASES AND W.
STABILITY SINTERING
OF PENETRATING PORES*
LIQUID
BEEREt
A model of grain edge porosity is presented which is equally applicable to liquid precipitates, to fission gas swelling in nuclear fuels and to powder compacts. The morphology of the pores is shown to depend on the ratio of their surface to grain boundary energies and their volumes. The stability of the porosity is shown to decrease with increasing dihedral angle. Pores with dihedral angles greater than 30” are shown to require a volume above a threshold value before long range interconnections are stable. The curvature of the porosity is also discussed with reference to its effect on diffusion prooesses. Liquid phases are shown to possess large driving forces for penetration and powder compacts to have large driving forces for sintering. The predictions of the theory are compared with experimental results obtained by diffusing liquid phases into polycrystals. UNE THEORIE
UNIFEE
DE LA STABILITE DES PHASES ET DES PORES DE FRITTAGE
LIQUIDES
PE~~TRANTES
On presante un modele de la porosit& au bord dun grain, applicable egalement aux pr&ipit& liquides, au gonflement par gaz de fission dans les combustibles nu&aires et aux poudres comprimees. On montre que Ia morphologie des pores depend de leur volume et du rapport de l’energies superficielle it l’energie intergranulaire. On montre que la stabilite de la porosite diminue lorsqu’augmente I’angie diedre. Les pores avec des angles diedres superieurs a 30 degres neoessitent un volume superieur B un certain seuil, avant que les connections Z%longue distance ne soient stables. On discute l’effet de la oourburo de la porosite sur lea processus de diffusion. On montre que les phases liquides presentent de grandes forces matrices pour la penetration et que les poudres comprimrfes ont de grandea forces matrices pour 1%frittage. Les previsions theoriques sont comparees a des resultats obtenus par diffusion de phases liquides dans de8 polycristaux. GEMEINSAMES
MODELL FUR STABILITAT VERMISCHTER PAASEN UND SINTERPOREN
FLUSSIGER
Es wird ein Model1 der Sinterporositiit (grain edge porosity) vorgeschlagen, das gleicherma~en auf fliissige Au~heid~gen, auf Spaltg~sohwellen in Ke~b~nnstoffen und auf Paver-kom~ktma~rialien anwendbar ist. Die Mo~hologi6 der Poren hangt vom Verhiiltnis von O~rfl~chenener~e zu Korngrenzen-energie und von ihrem Volumen ab. Die Stab%& der Poren nimmt mit zunehmendem dihedralen Winkel ab. Weitreichende Interkonnektionen zwischen Poren mit einem dihedralen Winkel von mehr als 30” sind nur oberhalv eines Volumenschwellwertes stabil. Die Krtimmung der Porositiit wird im Hinblick auf ihren EinfluD auf Diffusionsprozesse diskutiert. Flussige Phasen besitzen starke treibende Krafte fur Durchdringung und Pulverkompaktmaterialien fur Sintern. Die Vorhersagen der Theorie warden mit Ergebnissen aus Experimenten verglichen, in denen fliissige Phasen in Polykristalle eindiffundiert wurden. INTRODUCTION
been employed as powder sintering models. Cable@) constructed a model in which the porosity consisted of cylinders arranged along the edges of an idealized tetrak~idecahedrou grain. As a consequence the driving force for sintering decreased continually with increase in pore f&&ion. The grain edge structure may form when an agglomerate of particles bond together and the necks have grown to a considerable size. The resulting porosity is interlinked extensive and often unstable, sintering and eventually forming small isolated pores. The surface interfacial energy was greater than 1/i the ratio of surface to grain boundary energy in powder liquid phase always penetrated large distances forming compacts is higher than that for liquid precipitates. Recently theories of grain edge porosity have been a three dimensional network throughout the grains. When the ratio of energies was less than 2/i small developed to interpret swelling and gas release in volumes of the liquid phase penetrated only part pray nuclear fuels.(3-5) Gas created in the grains by the along the grain edge isolating the phase at the grain fission process can migrate to the grain bouud~ries forming large gas bubbles .@f The bubbles can grow corners. Interconnected grain edge structures have also and interlink along grain edges leading to the gas venting at the surface. The ratio of surface to grain * Received February 17, 1974; revised April 22, 1974. boundary energies is intermediate to those associated 7 Central Electricity Generating Board, Berkeley Nuclear with liquid metals and metal powder compacts. Laboratories, Berkeley, Gloucest,ershire, England.
The shapes adopted by gas pores or liquid phases situated along grain edges have been discussed in several fields of materials science. Smith(l) has reviewed liquid phases in metals. In particular the review covered two phase alloys made by heating the components above the solidus. On cooling the alloy was cold worked and annealed above the melting point of one of the phases. Recrystallization redistributed the liquid phase along the grain edges. When the ratio of the grain boundary to the liquid/
ACTA METALLURGICA,
VOL.
23, JANUARY
1976
131
132
ACTA
~E~ALLURGICA,
VOL.
23,
1975
The present work develops a model of grain edge porosity covering a wide range of surface to grain boundary energy ratios, and is applicable to the three types of porosity discussed. The model is based on energy considerations of the porosity and predicts shapes, surface curvatures and stability. The pore shapes derived are not cylindrical but have complex anticlastic or syn&stic curvature. The predictions are compared with observations of liquid precipitates produced by allowing liquid phases to penetrate the boundaries of polyerystals.
result of supe~aturation on cooling since at 930°C lead can dissolve about 30 atomic per cent copper. The structure produced by annealing at 900°C appeared identical to that produced at 930%. Reducing the annealing temperature further to 800°C still resulted in lead penetration and dendrite production but on a much smaller scale. A further series of anneals were performed at 900°C for times of 4,16 and 48 hr. Increased penetration was not observed but there was a considerable increase in grain size. Finally the volume of the lead was increased to EXPERIMENTAL about three times the volume of copper and the alloy Copper lead alloys were made by sealing com- annealed for l_Eihr at 900°C. The lead penetration mercially pure copper and lead in evacuated silica down the grain boundaries was similar to that obcapsules 3 mm dia bore and 40 mm long. The copper served for the smaller volume of lead but the copper was in the form of a rectangular prism 12 mm long x was wasted away from the outside. About a third by 2.5 mm wide and 1.6 mm thick. The volume of the volume of the copper had been dissolved by the lead lead was ~0.3 times that of the copper. Specimens forming dense copper dendrites in the lead surrounding of copper and lead were annealed for 1 hr at tempera- the copper polycrystal. tures of 930, 900 and 800°C. When making all the above copper lead alloys the When the copper was annealed at 93O”C, a tempera- molten lead touched the copper in the silica tube. If ture which is below the monote~tie ~mperature, the the metals did not touch but were in contact only copper maintained its polyGrystalline struoture. through their vapours then no penetration effect Figure 1 shows an optical micrograph of a polished was observed. section showing the lead at triple points. The lead Penetration was also studied in the IJO,/Al,O, penetrated down the grain boundaries from outside system. The uranium and aluminium oxides have a the copper. The copper displaced by the lead was simple eutectic phase diagramc5) with the eutectic removed by dissolving in the lead and subsequently temperature at ~1900°C. A cylindrical pellet of precipitating as dendrites outside the body of the uranium dioxide 10 mm long by 3.8 mm dia was copper. The dendrites were probably formed as a annealed for 24 hr in flowing hydrogen in an alumina crucible at ~1900°C. On removal from the furnace the uranium dioxide pellet appeared unchanged except for alumina crystals on its surface. The alumina crucible was darkened by contamination with UO,. Examination of a polished section, Fig. 2 showed alumina penetration down grain boundaries. Several hundred measurements were made of the dihedral angle of the alumina liquid phase. The mode of the distribution of angles was found to be about 15”. The UO, was fractured and scanning electron micrographs taken of the fracture surface Figs. 3(a and b). These clearly show the holes where the alumina had formed a long thin prism like phase along the grain edges. On fracturing the polycrystal the alumina had fallen out. Apart from a small area of contact the UO, and alumina had been in contact only through their vapours. THEORY
Fra. 1. An optical micrograph of a polished section through a copper-lead alloy annealed at 930% for 1 hr.
The equilibrium shape adopted by porosity or a liquid phase may be found from a consideration of its free energy. If we consider firstly the case of a liquid
BEERE:
STABILITY
OF
PENETRATING
LIQUID
energy
PHASES
AND
SINTERING
and grain boundary
PORES
energy
applies
133
equally
whether the shape is gas filled, liquid filled or empty. This is true in real situations when the change in volume
takes place slowly with rapid surface accomcurvature is equal modation, i.e. the surface everywhere and the dihedral angle at the boundary is a constant. problem
This assumption tractable
temperature processes. The equilibrium shape calculated on
the
by considering edges
grain, Fig. 4(a).
of
is necessary to make the
and is adequate
an
of grain
for
high
edge porosity
the porosity
idealized
many
is
to be situated
tetrakaidecahedron
The energy calculations
are based on
a unit of porosity situated on a grain corner, Fig. 4(b).
FIU. 2. An optical micrograph of a polished section through a UO, pellet annealed in an Al,O, environment for 24 hr at 1900°C.
precipitate
then the free energy E is given by E = AJ.5 -
(1)
f&Y,,
where A, is the area of the liquid-solid surface removed
by the liquid
of
phase and yp. is the surface
energy of the grain boundary. If the surfaces are allowed volume
interface
energy y8, A, is the area of grain boundary
of precipitate,
to relax at a constant
then the surfaces
adopt a shape of minimum energy. having the minimum energy satisfy
eventually
The surfaces the condition
i3#i3V = 0 where d+=dE--c-iv, and A is a Langrangian
(2)
multiplier.
The equilibrium
shapes of closed voids or open porosity may also be calculated from equation (2) if surface relaxation is much faster than volumetric
changes.
The equilibrium shape of closed gas bubbles may be found by incorporating in equation (1) a term for the work done on the enclosed gas of pressure p dE, = ys dA, -
ys dA, -
p dV,
(3)
where E, is the energy change in the bubble unclergoing a small change in shape and volume. When equation (1) is substituted into equation (2) it is seen that equations
(2 and 3) are identical
if p is put equal
to 1. Thus surfaces that satisfy equation (2) satisfy equation (3). It follows that an equilibrium shape calculated for a particular volume fraction, surface
(b) Fro. 3(a, b). Scanning electron micrographs of the fracture surface of UO, annealed in an Al,O, environment for 24 hr at 1900°C.
ACTA
134
METALLURGICA,
VOL.
23,
1975
FOG. 4(a). An idealized system of porosity extending round the edges of a tetrakeidecahedron grain. (b) A unit of porosity situated on a corner of the tet,rakaidecahedron.
The unit of porosity consists of an octahedron situated at the centre with four of the eight faces extended by Each extended face is surmounted by frustrums. four frustrums. The exposed faces of the frustrums and octahedron at a constant boundary.
are curved so that the surfaces meet
angle at the intersection
surface to grain boundary grain boundary
energies.
area removed
unit are calculated
The surface area,
and the volume
of the
from the five dimensions
al-a5,
Fig. 4(b) and the instantaneous
grain edge length 1 where the dihedral angle 19is given by cos 13= y,/2y,.
Fig. 4(a).
The height of each frustrum
small
make
to
smooth.
the
surface
is sufficiently
curvature
The approximations
hedrons
with the grain
The angle is chosen to satisfy the ratio of
Fra. 5(a). The predicted shape of two adjacent corner units when the dihedral angle is 15” and the volume fraction of precipitate is 2 per cent. (b) The predicted shape of two adjacent corner units when the dihedral angle is 90” and volume fraction of porosity is 10 per cent.
The precipitate
reasonably
made by placing
octa-
at the corners of a tetrakaidecahedron
are
is prismatic
Figure 5(b) shows the calculated
acceptable particularly at small pore volumes. The minimum energy of the configuration is found
is 90”, which effectively
by calculating
The porosity
on a computer
the value of the energy
according to equation (1). The volume is fixed and initially al-a5 are all equal to a value which satisfies the chosen small
volume.
sequential
culating
the
constant
volume.
The dimensions
additions
value
and
of a5 each Each
time
+a4
undergo
subtractions time
recal-
to maintain
a decrease
a
in energy
results from a change in length the new value is allowed to remain. After many changes in length the
along the grain edges and
joins smoothly at the corners. The prismatic sections remain stable as the volume fraction is reduced. shape of open poros-
ity of 10 per cent vol fraction.
connecting
is rounded
the
The dihedral
angle
removes the grain boundaries.
corners
in shape collapses
and the section
when
the volume
fraction is reduced below about 8 per cent. Figure
6 shows how the energy
changes
with volume
angles.
The energy
energy dimensionless When,
fraction is divided
of a corner unit
for different
dihedral
by ys12 to make the
and independent
of grain size.
8, is equal to 15” or 30” the values
of the
energy converges to the minimum for that volume and
energy are always less than zero. This is to be expected when the grain boundary is much more ener-
the values of al-a5 remain unaltered.
getic per unit area than the surface.
The procedure
is repeated for different volume fractions and different surface to grain boundary
energy ratios.
Since the change in shape occurs at constant volume no work is done on any gas or surfaces extraneous to the corner unit system. The shape of minimum energy is a true minimum irrespective of what the
Increasing
surface energy relative to the grain boundary increases the overall energy.
the
energy
At 8 = 45” the energy is
positive but goes through a maximum at ~12 per cent vol fraction. The curves for 0 > 30 are terminated on the left of Fig. 6 when the connecting porosity
porosity contains. Figure 5(a) shows the calculated shape of two adjacent corner units typical of a liquid
collapses. The curves are terminated on the right when the porosity consumes the grain boundaries. The boundaries are consumed when the porosity on
precipitat’e. The volume fraction of precipitate is taken as 2 per cent and the dihedral angle 15’,
opposite faces of the square grain faces, meet in the middle of the face.
Fig. 4(a)
BEERE:
STABILITY
OF
PENETRATING
LIQUID
PHASES
AND
SINTERING
PORES
135
The result is 27,/r where 2/r is the curvature of the hemispherical caps.) The curvature of the grain edge porosity is shown in Fig. 7. The curvature is defined in the sense that a spherical cavity has positive surface curvature whereas a spherical solid object has negative curvature. When the dihedral angle is 15’ the curvature tends to minus infinity as the volume fraction tends to zero. The surface of the porosity connecting grain corners can be thought of as strips of surface on the inside of three parallel t,ouching cylinders. As the diameter of the cylinders tends to zero so the curvature tends to minus infinity. This result holds true for 8 < 30”. At 30” exactly the curvature goes through the origin. This is because a section through the precipitate perpendicular to t,he grain 0 .l .2 3 L ,5 6 edge is an equilateral triangle with straight sides. VOLUME FRACTION The precipitate along the grain edge is prism like Fru. 6. The dimensionless energy of a corner unit of and so at small volumes the surfaces are plane and the porosity versus volume fraction. eurvatur~ is zero. In practical terms the boundaries The minimum energy for a particular volume is the are never orientated equidistantly at 120” intervals. energy of a pore with surfaces everywhere at equilib- As a result the section through the precipitate does rium. The pore may not be at equilibrium with re- not have straight sides and the conditions for zero spect to a change in volume. The tendency to shrink curvature cannot be met. Above a dihedral angle of 30” the tendency is for or grow may be found by considering the change in the curvature to become infinite as the volume deminimum energy with change in volume. This creases but instability collapses the interconnections allows a calculation of the surface potential which at a finite voIume fraction porosity. may then be compared with the potential at the When 0 = 45” the curvature is negative for volume grain boundaries to determine the direction of fractions above 12 per cent. The surface curvature at matter flow. a point can be defined in terms of the sum of two The chemical potential of atoms on the surface of a perpendicular curvatures l/r1 and l/r,, Fig. 4(b). void is given by ,u* = iTEj&, where n is the number of When 0 = 45”, I/ rl is positive and l/r, is negative. surface atoms. Putting Sn c= -SVQ, where Q is the Increasing the pore volume decreases l/r, and l/r,. atomic volume then ,uu,= -( aE/aV)Q. The chemical At 12 per cent vol fraction the sum becomes negative. pot’ential in the presence of a gas is ,u, = - (aE/a V + p)Q - --(aEJaV)n. The surface curvature X on a void, can be found from the Gibbs-Thompson relatio~hip ;thS= -y&Q and is given by Ky, = i3EjElaV.
(4)
The same result may be derived for a gas filled pore by finding the gas pressure at equi~brium with the surface tension restraint. Then dE,, equation (3), is zero and (5) But the bracketed term is equal to E and p = aEEfi3V. If the external restraints are zero then p = KY, and Kys = iW@V. (The curvature of a lenticular bubble on a grain boundary may be calculated exactly by equation (4).
SURFACE FIG.
7. The
CURVATURE
i l/r, + 1/rz)
dimensionless curvature versus volume fraction of grain edge porosity.
136
ACTA DISCUSSION
-Liquid precipitates
METALLURGICA,
VOL.
23,
1975
ably darkened by UO,. The UO, pellet became covered in small alumina crystals. The small alumina crystals presumably absorb UO, to become a liquid phase which subsequently penetrates the boundaries. Measurements of changes in specimen length were inconclusive owing to surface evaporation and surface growing. Thus t(he results do not determine if the UO, displaced by the alumina diffuses to the grain boundaries or evaporates from the system. The fractographs of the UO, pellet, Fig. 3(a), show the liquid phase occupying the grain edges in a manner similar to the idealized model Fig. 4(a). The observed shape of the liquid phase Fig. 3(b) matches the thin prismatic morphology predicted by the calculations Fig. 5(a). The computer models also agree with previous work(lO) regarding the stability of the precipitate between grain corners. The calculation show that when 0 < 30” the interconnections between corners remain intact as the precipitate volume tends to zero. The model also shows that stability exists between neighboring sections of precipitates. If two regions of precipitate have different volume fractions then their surfaces have different curvatures. Since the potential of surface atoms pu,is equal to -Ky8Q the atoms on the surface of smaller volume represented by point “a” Fig. 7, have a higher potential than atoms on the surface of larger volume represented by “b”. Atoms diffuse from a to b tending to equal out the difference in sizes.
The dihedral angle of liquid lead in copper has been reported@) as 30” at 800°C falling to zero at 950°C. Reference to Fig. 6 shows that when f3 5 30” increasing the volume of liquid precipitates lowers the free energy and hence the tendency will be for the volume of precipitate to increase. At very small volume fractions of precipitate for which 0 < 30 the liquid/solid interfacial curvature tends to minus infinity limited by the atomic diameter. This behaviour is shown in Fig. 7 for 8 = 15”. The equilibrium concentration of dissolved copper immediately above a curved surface is enhanced by an amount where k is Boltzmanns constant exp (--Ky,WT) and T is absolute temperature. When the lead first penetrates the boundaries the very large negative curvature greatly enhances the concentration of dissolved copper. The copper is dissolved more rapidly from the interior grain edges than the exterior surface were the curvature is less negative. At 900’ molten lead can dissolve 25 atomic per cent copper.(g) If the copper is in contact with 0.3 times its own volume in lead, then the lead can dissolve 3 per cent by volume of the copper polycrystal. This is consistent with the observed penetration, Fig. 1, which showed that no more than a few per cent of copper dissolved after long anneals. Increasing the volume of lead to 3 times that of the copper allowed the lead to dissolve 30 per cent of the copper. The extra copper is not all dissolved from IRRADIATED UO, the interior of the copper polycrystal however beThe dihedral angle of fission gas porosity in irracause the curvature in the interior increases with volume fraction and would exceed the curvature on diated UOz has been reported as 45”.(11) The calthe exterior. The exterior boundaries become grooved culations predict that porosity having this dihedral and each surface grain takes on a negative curvature. angle will form connections between grain corners If it is assumed that the exterior surface has spherical only when the volume fraction is greater than 4 per curvature then the curvature is about --1.4/l. At cent. Gas created by the fission process can diffuse to 900°C the dihedral angle 8 is about 25’(*) and the the grain boundaries where large bubbles are formed. interior curvature tends to minus infinity as the vol- In certain temperature regimes coalescence of bubbles ume fraction tends to zero. Interpolating between on the grain edges can create a connection. Long the curvatures for 15” and 30” Fig. 7 it can be in- range linkage o,f the connections occurs when about and gas is ferred that the interior curvature equals --1.4/l at half the grain edges are connected quite a small volume fraction of precipitate. In- released from the fue1.(13) The equilibrium gas creasing the volume fraction of precipitate further pressure in the tunnels oan be found from the surface increases the curvature resulting in a migration of curvature. Thus the calculations enable a more copper atoms from the exterior to the interior sur- accurate estimate than has previously been possible faces. Hence copper is dissolved only from the of the incubation time to gas release. exterior. The energy curves Pig. 6 show that when 8 = 45” The UOJAl,O, system also showed penetration of the dimensionless energy of the tunnels E/y,P dealumina rich liquid phase down the UO, grain boun- creases with increase in volume fraction porosity daries. At 1900°C the two components interacted via above 12 per cent. This does not mean that the larger their vapours and the alumina crucible was consider- pore sizes will grow however, because material
BEERE:
STABILITY
OF
PENETRATING
removed by the porosity is accommodated on the grain bounda~es increasing the grain edge length I. Plotting ~/y~~2 vs volume fraction, where lo is the value of 1 when the pore fraction is zero, shows that, the energy increases with increase in porosity above 12 per cent, even when the surface curvature is negative. This result is confirmed by considering the energy balance when atoms are removed from the boundary. If the boundary is a perfect source and sink for vacancies, then for every local value of the grain boundary atom potential p,,, there corresponds a 1ocaI normal stress G given by G = &LX When there are no external forces the sum of all the forces integrated over the boundary must equal the surface tension forces at the intersection with the porosity. Integrating over the boundary
LIQUID
PHASES
AND
SINTERING
PORES
137
pattern with many boundaries intersecting a cavity. As the necks between particles become larger relative to the particle size it is expected that the porosity becomes more regular and is situated along grain edges. The dihedral angles in powder compacts are often greater than 45’. Figure 7 shows that the grain edge geometry can form at about 30 per cent porosity when 8 = 45” screwing to slightly over 60 per cent when 8 = 90’. Above these porosities the cavities have relatively more grain boundaries intersecting them, the surface curvature is more negative and the cavities may show little tendency to shrink.{14 G) Once the grain edge structure develops, however there is a thermodyuamic tendency to sinter. The effect different green densities have on sintering rates can be inferred from Fig. 7. Point e represents a Ly, sin 8 = d dA, (6) powder having a dihedral angle of 75” which after the s initial neck growth stage has too muoh porosity to form a grain edge structure. S~te~ng in this case will where L is the length of the intersection with the be slow. Increasing the green density may enable a porosity. grain edge structure to develop more readily resulting If there is equilibrium on pore surfaces and boundin rapid sintering. This case is represented by point f. ary, then there are no atom fluxes, pa is not a funcBehavior of this type has been observed in UO, tion of position and pa = p,. Su~tituting for G in powders.tlq) When the green density was greater than equation (6) and integrat~g 50 per cent the powder sintered much more quickly than when the green density was less than 50 per ,us = y&Q sin O/A, (7) cent. The low density powder represented by point e, where A is the area of the boundary, Substituting Fig. 7 may sinter more readily if grain growth removes for the curvature from the relation ps = -Ky&2 the the boundaries from some of the pores thus increasing surface curvature of porosity that neither shrinks nor the surface curvature to point g. These pores will now grows is negative and is given by the equation sinter by diflYnsionto other pores still intersected by K = --L sin S/A. A separate calculation of this boundaries or by diffusion to nearby boundaries. curvature shows that the pores always tend to siuter The curvature of the pores shrinking represented by when the dihedral angle 8 is 45’ 60’ I 75’ or 90”. point g, Pig. 7, descends the @ = 90 curve until their Indeed when the volume fraction of porosity is high size is equivalent to about 8 per cent porosity at and the surface curvature is negative there is a large which point the connecting links collapse isolating the driving force for sin&ring. This foroe results from the pore in the grain. This behavior has been observed surface tension forces acting on the small areas of in low density UO, powder compacts.(14) grain boundary. CONCLUSIONS The pores were also shown to be unstable with reThe model of grain edge porosity outlined is approspe~ttoneighboriugre~onsof slightly different volume fractions. Two pore volumes and their associate priate when discussing liquid preeipita~s, fission gas surfaoe curvature are represented by the points c and swelling and powder compacts. The morphology d, Fig. 7. The atom potential on the larger pore c is of the porosity is dependent on its contents only greater than that on the smaller pore d. Atoms will through the influence they have on the ratio ofsurface tend to migrate from c to d increasing the size of the to grain boundary energies. The model suooessfulIy predicts the shape of liquid precipitates having a larger pore at the expense of the smaller. dihedral angle of ~15’. Also the model demonstrates POWDER COMPACTS the long and short range stability of the precipitates. Interconnected porosity in powder compacts results Porosity with dihedral angles greater than 30” is from the formation of necks between adjacent shown to require a threshold volume before linkage particles. Initially the porosity may have no regular along the grain edges is locally stable. It is also shown
ACTA
138
however
that
this porosity
METALLURGICA,
is long range unstable,
a
large pore tending to grow at the expense of a smaller neighbor.
Finally
the
sintering
behavior
of UO,
powders may be explained by the changes in pore curvature associated with grain growth and change in green density. ACKNOWLEDGEMENT
This paper is published by permission of the Central Electricity
Generating
Board.
23,
1976
4. W. BEERE, Int. Conf. on Physical Elements. Berkelev. (19731.
Met. of Reactor
Fuel
anyi. A. ~L’TJRNBULL, ibid. 5. M. 0. T&ER 6. G. L. REYNOLDS and G. H. BANNISTER, J. Mat. Sci.
84 (1970).
5,
7. L. F. EPSTEIN and W. H. HOWLAND. J. Amer. Ceram. Sot. 36. 334 (19531. 185, 8. K. K. &ENY& and C. S. SMITH, Trans. A.I.M.E. 762 (1949). 9. 0. J. KLEPPA and J. A. WEIL, J. Am. Chem. Sot., 73, 4848 (1951). 176. 15 (1948). 10. C. S. SMITH. Trans. A.I.M.E. 11. G. L. REYNOLDS, W. B. BEERE and @. T.’ SAWBRIDOE, J. Nud. Mater. 41, 112 (1971). 12. H. L. FRISCH, J. M. HAMMERSLEY and D. J. A. WELSH,
Phye. Rev. 126, 949 (1962).
REFERENCES 1. C. S. SMITH, Met. Rev. 9, 1 (1964). 2. R. L. COBLE, J. A&. Phys. 32, 787 (1961). 3. W. B. BEERE and G. L. REYNOLDS, J. Nucl. 51 (1973).
VOL.
Mater.
47,
8, 1717 (1973). 14. W. BEERE,J. Mat.Sci. 15. W. D. KINGERY and B. FRAN(:OIS, Sintering and Related Phenomena, edited by G. C. KUCZYNSKI, p. 471. Gordon & Breach, New York (1967). 16. W. D. KINGERY and B. FRANCOIS, ibid. p. 499 (1967).
THEORY OF THE PHASES AND W.
STABILITY SINTERING
OF PENETRATING PORES*
LIQUID
BEEREt
A model of grain edge porosity is presented which is equally applicable to liquid precipitates, to fission gas swelling in nuclear fuels and to powder compacts. The morphology of the pores is shown to depend on the ratio of their surface to grain boundary energies and their volumes. The stability of the porosity is shown to decrease with increasing dihedral angle. Pores with dihedral angles greater than 30” are shown to require a volume above a threshold value before long range interconnections are stable. The curvature of the porosity is also discussed with reference to its effect on diffusion prooesses. Liquid phases are shown to possess large driving forces for penetration and powder compacts to have large driving forces for sintering. The predictions of the theory are compared with experimental results obtained by diffusing liquid phases into polycrystals. UNE THEORIE
UNIFEE
DE LA STABILITE DES PHASES ET DES PORES DE FRITTAGE
LIQUIDES
PE~~TRANTES
On presante un modele de la porosit& au bord dun grain, applicable egalement aux pr&ipit& liquides, au gonflement par gaz de fission dans les combustibles nu&aires et aux poudres comprimees. On montre que Ia morphologie des pores depend de leur volume et du rapport de l’energies superficielle it l’energie intergranulaire. On montre que la stabilite de la porosite diminue lorsqu’augmente I’angie diedre. Les pores avec des angles diedres superieurs a 30 degres neoessitent un volume superieur B un certain seuil, avant que les connections Z%longue distance ne soient stables. On discute l’effet de la oourburo de la porosite sur lea processus de diffusion. On montre que les phases liquides presentent de grandes forces matrices pour la penetration et que les poudres comprimrfes ont de grandea forces matrices pour 1%frittage. Les previsions theoriques sont comparees a des resultats obtenus par diffusion de phases liquides dans de8 polycristaux. GEMEINSAMES
MODELL FUR STABILITAT VERMISCHTER PAASEN UND SINTERPOREN
FLUSSIGER
Es wird ein Model1 der Sinterporositiit (grain edge porosity) vorgeschlagen, das gleicherma~en auf fliissige Au~heid~gen, auf Spaltg~sohwellen in Ke~b~nnstoffen und auf Paver-kom~ktma~rialien anwendbar ist. Die Mo~hologi6 der Poren hangt vom Verhiiltnis von O~rfl~chenener~e zu Korngrenzen-energie und von ihrem Volumen ab. Die Stab%& der Poren nimmt mit zunehmendem dihedralen Winkel ab. Weitreichende Interkonnektionen zwischen Poren mit einem dihedralen Winkel von mehr als 30” sind nur oberhalv eines Volumenschwellwertes stabil. Die Krtimmung der Porositiit wird im Hinblick auf ihren EinfluD auf Diffusionsprozesse diskutiert. Flussige Phasen besitzen starke treibende Krafte fur Durchdringung und Pulverkompaktmaterialien fur Sintern. Die Vorhersagen der Theorie warden mit Ergebnissen aus Experimenten verglichen, in denen fliissige Phasen in Polykristalle eindiffundiert wurden. INTRODUCTION
been employed as powder sintering models. Cable@) constructed a model in which the porosity consisted of cylinders arranged along the edges of an idealized tetrak~idecahedrou grain. As a consequence the driving force for sintering decreased continually with increase in pore f&&ion. The grain edge structure may form when an agglomerate of particles bond together and the necks have grown to a considerable size. The resulting porosity is interlinked extensive and often unstable, sintering and eventually forming small isolated pores. The surface interfacial energy was greater than 1/i the ratio of surface to grain boundary energy in powder liquid phase always penetrated large distances forming compacts is higher than that for liquid precipitates. Recently theories of grain edge porosity have been a three dimensional network throughout the grains. When the ratio of energies was less than 2/i small developed to interpret swelling and gas release in volumes of the liquid phase penetrated only part pray nuclear fuels.(3-5) Gas created in the grains by the along the grain edge isolating the phase at the grain fission process can migrate to the grain bouud~ries forming large gas bubbles .@f The bubbles can grow corners. Interconnected grain edge structures have also and interlink along grain edges leading to the gas venting at the surface. The ratio of surface to grain * Received February 17, 1974; revised April 22, 1974. boundary energies is intermediate to those associated 7 Central Electricity Generating Board, Berkeley Nuclear with liquid metals and metal powder compacts. Laboratories, Berkeley, Gloucest,ershire, England.
The shapes adopted by gas pores or liquid phases situated along grain edges have been discussed in several fields of materials science. Smith(l) has reviewed liquid phases in metals. In particular the review covered two phase alloys made by heating the components above the solidus. On cooling the alloy was cold worked and annealed above the melting point of one of the phases. Recrystallization redistributed the liquid phase along the grain edges. When the ratio of the grain boundary to the liquid/
ACTA METALLURGICA,
VOL.
23, JANUARY
1976
131
132
ACTA
~E~ALLURGICA,
VOL.
23,
1975
The present work develops a model of grain edge porosity covering a wide range of surface to grain boundary energy ratios, and is applicable to the three types of porosity discussed. The model is based on energy considerations of the porosity and predicts shapes, surface curvatures and stability. The pore shapes derived are not cylindrical but have complex anticlastic or syn&stic curvature. The predictions are compared with observations of liquid precipitates produced by allowing liquid phases to penetrate the boundaries of polyerystals.
result of supe~aturation on cooling since at 930°C lead can dissolve about 30 atomic per cent copper. The structure produced by annealing at 900°C appeared identical to that produced at 930%. Reducing the annealing temperature further to 800°C still resulted in lead penetration and dendrite production but on a much smaller scale. A further series of anneals were performed at 900°C for times of 4,16 and 48 hr. Increased penetration was not observed but there was a considerable increase in grain size. Finally the volume of the lead was increased to EXPERIMENTAL about three times the volume of copper and the alloy Copper lead alloys were made by sealing com- annealed for l_Eihr at 900°C. The lead penetration mercially pure copper and lead in evacuated silica down the grain boundaries was similar to that obcapsules 3 mm dia bore and 40 mm long. The copper served for the smaller volume of lead but the copper was in the form of a rectangular prism 12 mm long x was wasted away from the outside. About a third by 2.5 mm wide and 1.6 mm thick. The volume of the volume of the copper had been dissolved by the lead lead was ~0.3 times that of the copper. Specimens forming dense copper dendrites in the lead surrounding of copper and lead were annealed for 1 hr at tempera- the copper polycrystal. tures of 930, 900 and 800°C. When making all the above copper lead alloys the When the copper was annealed at 93O”C, a tempera- molten lead touched the copper in the silica tube. If ture which is below the monote~tie ~mperature, the the metals did not touch but were in contact only copper maintained its polyGrystalline struoture. through their vapours then no penetration effect Figure 1 shows an optical micrograph of a polished was observed. section showing the lead at triple points. The lead Penetration was also studied in the IJO,/Al,O, penetrated down the grain boundaries from outside system. The uranium and aluminium oxides have a the copper. The copper displaced by the lead was simple eutectic phase diagramc5) with the eutectic removed by dissolving in the lead and subsequently temperature at ~1900°C. A cylindrical pellet of precipitating as dendrites outside the body of the uranium dioxide 10 mm long by 3.8 mm dia was copper. The dendrites were probably formed as a annealed for 24 hr in flowing hydrogen in an alumina crucible at ~1900°C. On removal from the furnace the uranium dioxide pellet appeared unchanged except for alumina crystals on its surface. The alumina crucible was darkened by contamination with UO,. Examination of a polished section, Fig. 2 showed alumina penetration down grain boundaries. Several hundred measurements were made of the dihedral angle of the alumina liquid phase. The mode of the distribution of angles was found to be about 15”. The UO, was fractured and scanning electron micrographs taken of the fracture surface Figs. 3(a and b). These clearly show the holes where the alumina had formed a long thin prism like phase along the grain edges. On fracturing the polycrystal the alumina had fallen out. Apart from a small area of contact the UO, and alumina had been in contact only through their vapours. THEORY
Fra. 1. An optical micrograph of a polished section through a copper-lead alloy annealed at 930% for 1 hr.
The equilibrium shape adopted by porosity or a liquid phase may be found from a consideration of its free energy. If we consider firstly the case of a liquid
BEERE:
STABILITY
OF
PENETRATING
LIQUID
energy
PHASES
AND
SINTERING
and grain boundary
PORES
energy
applies
133
equally
whether the shape is gas filled, liquid filled or empty. This is true in real situations when the change in volume
takes place slowly with rapid surface accomcurvature is equal modation, i.e. the surface everywhere and the dihedral angle at the boundary is a constant. problem
This assumption tractable
temperature processes. The equilibrium shape calculated on
the
by considering edges
grain, Fig. 4(a).
of
is necessary to make the
and is adequate
an
of grain
for
high
edge porosity
the porosity
idealized
many
is
to be situated
tetrakaidecahedron
The energy calculations
are based on
a unit of porosity situated on a grain corner, Fig. 4(b).
FIU. 2. An optical micrograph of a polished section through a UO, pellet annealed in an Al,O, environment for 24 hr at 1900°C.
precipitate
then the free energy E is given by E = AJ.5 -
(1)
f&Y,,
where A, is the area of the liquid-solid surface removed
by the liquid
of
phase and yp. is the surface
energy of the grain boundary. If the surfaces are allowed volume
interface
energy y8, A, is the area of grain boundary
of precipitate,
to relax at a constant
then the surfaces
adopt a shape of minimum energy. having the minimum energy satisfy
eventually
The surfaces the condition
i3#i3V = 0 where d+=dE--c-iv, and A is a Langrangian
(2)
multiplier.
The equilibrium
shapes of closed voids or open porosity may also be calculated from equation (2) if surface relaxation is much faster than volumetric
changes.
The equilibrium shape of closed gas bubbles may be found by incorporating in equation (1) a term for the work done on the enclosed gas of pressure p dE, = ys dA, -
ys dA, -
p dV,
(3)
where E, is the energy change in the bubble unclergoing a small change in shape and volume. When equation (1) is substituted into equation (2) it is seen that equations
(2 and 3) are identical
if p is put equal
to 1. Thus surfaces that satisfy equation (2) satisfy equation (3). It follows that an equilibrium shape calculated for a particular volume fraction, surface
(b) Fro. 3(a, b). Scanning electron micrographs of the fracture surface of UO, annealed in an Al,O, environment for 24 hr at 1900°C.
ACTA
134
METALLURGICA,
VOL.
23,
1975
FOG. 4(a). An idealized system of porosity extending round the edges of a tetrakeidecahedron grain. (b) A unit of porosity situated on a corner of the tet,rakaidecahedron.
The unit of porosity consists of an octahedron situated at the centre with four of the eight faces extended by Each extended face is surmounted by frustrums. four frustrums. The exposed faces of the frustrums and octahedron at a constant boundary.
are curved so that the surfaces meet
angle at the intersection
surface to grain boundary grain boundary
energies.
area removed
unit are calculated
The surface area,
and the volume
of the
from the five dimensions
al-a5,
Fig. 4(b) and the instantaneous
grain edge length 1 where the dihedral angle 19is given by cos 13= y,/2y,.
Fig. 4(a).
The height of each frustrum
small
make
to
smooth.
the
surface
is sufficiently
curvature
The approximations
hedrons
with the grain
The angle is chosen to satisfy the ratio of
Fra. 5(a). The predicted shape of two adjacent corner units when the dihedral angle is 15” and the volume fraction of precipitate is 2 per cent. (b) The predicted shape of two adjacent corner units when the dihedral angle is 90” and volume fraction of porosity is 10 per cent.
The precipitate
reasonably
made by placing
octa-
at the corners of a tetrakaidecahedron
are
is prismatic
Figure 5(b) shows the calculated
acceptable particularly at small pore volumes. The minimum energy of the configuration is found
is 90”, which effectively
by calculating
The porosity
on a computer
the value of the energy
according to equation (1). The volume is fixed and initially al-a5 are all equal to a value which satisfies the chosen small
volume.
sequential
culating
the
constant
volume.
The dimensions
additions
value
and
of a5 each Each
time
+a4
undergo
subtractions time
recal-
to maintain
a decrease
a
in energy
results from a change in length the new value is allowed to remain. After many changes in length the
along the grain edges and
joins smoothly at the corners. The prismatic sections remain stable as the volume fraction is reduced. shape of open poros-
ity of 10 per cent vol fraction.
connecting
is rounded
the
The dihedral
angle
removes the grain boundaries.
corners
in shape collapses
and the section
when
the volume
fraction is reduced below about 8 per cent. Figure
6 shows how the energy
changes
with volume
angles.
The energy
energy dimensionless When,
fraction is divided
of a corner unit
for different
dihedral
by ys12 to make the
and independent
of grain size.
8, is equal to 15” or 30” the values
of the
energy converges to the minimum for that volume and
energy are always less than zero. This is to be expected when the grain boundary is much more ener-
the values of al-a5 remain unaltered.
getic per unit area than the surface.
The procedure
is repeated for different volume fractions and different surface to grain boundary
energy ratios.
Since the change in shape occurs at constant volume no work is done on any gas or surfaces extraneous to the corner unit system. The shape of minimum energy is a true minimum irrespective of what the
Increasing
surface energy relative to the grain boundary increases the overall energy.
the
energy
At 8 = 45” the energy is
positive but goes through a maximum at ~12 per cent vol fraction. The curves for 0 > 30 are terminated on the left of Fig. 6 when the connecting porosity
porosity contains. Figure 5(a) shows the calculated shape of two adjacent corner units typical of a liquid
collapses. The curves are terminated on the right when the porosity consumes the grain boundaries. The boundaries are consumed when the porosity on
precipitat’e. The volume fraction of precipitate is taken as 2 per cent and the dihedral angle 15’,
opposite faces of the square grain faces, meet in the middle of the face.
Fig. 4(a)
BEERE:
STABILITY
OF
PENETRATING
LIQUID
PHASES
AND
SINTERING
PORES
135
The result is 27,/r where 2/r is the curvature of the hemispherical caps.) The curvature of the grain edge porosity is shown in Fig. 7. The curvature is defined in the sense that a spherical cavity has positive surface curvature whereas a spherical solid object has negative curvature. When the dihedral angle is 15’ the curvature tends to minus infinity as the volume fraction tends to zero. The surface of the porosity connecting grain corners can be thought of as strips of surface on the inside of three parallel t,ouching cylinders. As the diameter of the cylinders tends to zero so the curvature tends to minus infinity. This result holds true for 8 < 30”. At 30” exactly the curvature goes through the origin. This is because a section through the precipitate perpendicular to t,he grain 0 .l .2 3 L ,5 6 edge is an equilateral triangle with straight sides. VOLUME FRACTION The precipitate along the grain edge is prism like Fru. 6. The dimensionless energy of a corner unit of and so at small volumes the surfaces are plane and the porosity versus volume fraction. eurvatur~ is zero. In practical terms the boundaries The minimum energy for a particular volume is the are never orientated equidistantly at 120” intervals. energy of a pore with surfaces everywhere at equilib- As a result the section through the precipitate does rium. The pore may not be at equilibrium with re- not have straight sides and the conditions for zero spect to a change in volume. The tendency to shrink curvature cannot be met. Above a dihedral angle of 30” the tendency is for or grow may be found by considering the change in the curvature to become infinite as the volume deminimum energy with change in volume. This creases but instability collapses the interconnections allows a calculation of the surface potential which at a finite voIume fraction porosity. may then be compared with the potential at the When 0 = 45” the curvature is negative for volume grain boundaries to determine the direction of fractions above 12 per cent. The surface curvature at matter flow. a point can be defined in terms of the sum of two The chemical potential of atoms on the surface of a perpendicular curvatures l/r1 and l/r,, Fig. 4(b). void is given by ,u* = iTEj&, where n is the number of When 0 = 45”, I/ rl is positive and l/r, is negative. surface atoms. Putting Sn c= -SVQ, where Q is the Increasing the pore volume decreases l/r, and l/r,. atomic volume then ,uu,= -( aE/aV)Q. The chemical At 12 per cent vol fraction the sum becomes negative. pot’ential in the presence of a gas is ,u, = - (aE/a V + p)Q - --(aEJaV)n. The surface curvature X on a void, can be found from the Gibbs-Thompson relatio~hip ;thS= -y&Q and is given by Ky, = i3EjElaV.
(4)
The same result may be derived for a gas filled pore by finding the gas pressure at equi~brium with the surface tension restraint. Then dE,, equation (3), is zero and (5) But the bracketed term is equal to E and p = aEEfi3V. If the external restraints are zero then p = KY, and Kys = iW@V. (The curvature of a lenticular bubble on a grain boundary may be calculated exactly by equation (4).
SURFACE FIG.
7. The
CURVATURE
i l/r, + 1/rz)
dimensionless curvature versus volume fraction of grain edge porosity.
136
ACTA DISCUSSION
-Liquid precipitates
METALLURGICA,
VOL.
23,
1975
ably darkened by UO,. The UO, pellet became covered in small alumina crystals. The small alumina crystals presumably absorb UO, to become a liquid phase which subsequently penetrates the boundaries. Measurements of changes in specimen length were inconclusive owing to surface evaporation and surface growing. Thus t(he results do not determine if the UO, displaced by the alumina diffuses to the grain boundaries or evaporates from the system. The fractographs of the UO, pellet, Fig. 3(a), show the liquid phase occupying the grain edges in a manner similar to the idealized model Fig. 4(a). The observed shape of the liquid phase Fig. 3(b) matches the thin prismatic morphology predicted by the calculations Fig. 5(a). The computer models also agree with previous work(lO) regarding the stability of the precipitate between grain corners. The calculation show that when 0 < 30” the interconnections between corners remain intact as the precipitate volume tends to zero. The model also shows that stability exists between neighboring sections of precipitates. If two regions of precipitate have different volume fractions then their surfaces have different curvatures. Since the potential of surface atoms pu,is equal to -Ky8Q the atoms on the surface of smaller volume represented by point “a” Fig. 7, have a higher potential than atoms on the surface of larger volume represented by “b”. Atoms diffuse from a to b tending to equal out the difference in sizes.
The dihedral angle of liquid lead in copper has been reported@) as 30” at 800°C falling to zero at 950°C. Reference to Fig. 6 shows that when f3 5 30” increasing the volume of liquid precipitates lowers the free energy and hence the tendency will be for the volume of precipitate to increase. At very small volume fractions of precipitate for which 0 < 30 the liquid/solid interfacial curvature tends to minus infinity limited by the atomic diameter. This behaviour is shown in Fig. 7 for 8 = 15”. The equilibrium concentration of dissolved copper immediately above a curved surface is enhanced by an amount where k is Boltzmanns constant exp (--Ky,WT) and T is absolute temperature. When the lead first penetrates the boundaries the very large negative curvature greatly enhances the concentration of dissolved copper. The copper is dissolved more rapidly from the interior grain edges than the exterior surface were the curvature is less negative. At 900’ molten lead can dissolve 25 atomic per cent copper.(g) If the copper is in contact with 0.3 times its own volume in lead, then the lead can dissolve 3 per cent by volume of the copper polycrystal. This is consistent with the observed penetration, Fig. 1, which showed that no more than a few per cent of copper dissolved after long anneals. Increasing the volume of lead to 3 times that of the copper allowed the lead to dissolve 30 per cent of the copper. The extra copper is not all dissolved from IRRADIATED UO, the interior of the copper polycrystal however beThe dihedral angle of fission gas porosity in irracause the curvature in the interior increases with volume fraction and would exceed the curvature on diated UOz has been reported as 45”.(11) The calthe exterior. The exterior boundaries become grooved culations predict that porosity having this dihedral and each surface grain takes on a negative curvature. angle will form connections between grain corners If it is assumed that the exterior surface has spherical only when the volume fraction is greater than 4 per curvature then the curvature is about --1.4/l. At cent. Gas created by the fission process can diffuse to 900°C the dihedral angle 8 is about 25’(*) and the the grain boundaries where large bubbles are formed. interior curvature tends to minus infinity as the vol- In certain temperature regimes coalescence of bubbles ume fraction tends to zero. Interpolating between on the grain edges can create a connection. Long the curvatures for 15” and 30” Fig. 7 it can be in- range linkage o,f the connections occurs when about and gas is ferred that the interior curvature equals --1.4/l at half the grain edges are connected quite a small volume fraction of precipitate. In- released from the fue1.(13) The equilibrium gas creasing the volume fraction of precipitate further pressure in the tunnels oan be found from the surface increases the curvature resulting in a migration of curvature. Thus the calculations enable a more copper atoms from the exterior to the interior sur- accurate estimate than has previously been possible faces. Hence copper is dissolved only from the of the incubation time to gas release. exterior. The energy curves Pig. 6 show that when 8 = 45” The UOJAl,O, system also showed penetration of the dimensionless energy of the tunnels E/y,P dealumina rich liquid phase down the UO, grain boun- creases with increase in volume fraction porosity daries. At 1900°C the two components interacted via above 12 per cent. This does not mean that the larger their vapours and the alumina crucible was consider- pore sizes will grow however, because material
BEERE:
STABILITY
OF
PENETRATING
removed by the porosity is accommodated on the grain bounda~es increasing the grain edge length I. Plotting ~/y~~2 vs volume fraction, where lo is the value of 1 when the pore fraction is zero, shows that, the energy increases with increase in porosity above 12 per cent, even when the surface curvature is negative. This result is confirmed by considering the energy balance when atoms are removed from the boundary. If the boundary is a perfect source and sink for vacancies, then for every local value of the grain boundary atom potential p,,, there corresponds a 1ocaI normal stress G given by G = &LX When there are no external forces the sum of all the forces integrated over the boundary must equal the surface tension forces at the intersection with the porosity. Integrating over the boundary
LIQUID
PHASES
AND
SINTERING
PORES
137
pattern with many boundaries intersecting a cavity. As the necks between particles become larger relative to the particle size it is expected that the porosity becomes more regular and is situated along grain edges. The dihedral angles in powder compacts are often greater than 45’. Figure 7 shows that the grain edge geometry can form at about 30 per cent porosity when 8 = 45” screwing to slightly over 60 per cent when 8 = 90’. Above these porosities the cavities have relatively more grain boundaries intersecting them, the surface curvature is more negative and the cavities may show little tendency to shrink.{14 G) Once the grain edge structure develops, however there is a thermodyuamic tendency to sinter. The effect different green densities have on sintering rates can be inferred from Fig. 7. Point e represents a Ly, sin 8 = d dA, (6) powder having a dihedral angle of 75” which after the s initial neck growth stage has too muoh porosity to form a grain edge structure. S~te~ng in this case will where L is the length of the intersection with the be slow. Increasing the green density may enable a porosity. grain edge structure to develop more readily resulting If there is equilibrium on pore surfaces and boundin rapid sintering. This case is represented by point f. ary, then there are no atom fluxes, pa is not a funcBehavior of this type has been observed in UO, tion of position and pa = p,. Su~tituting for G in powders.tlq) When the green density was greater than equation (6) and integrat~g 50 per cent the powder sintered much more quickly than when the green density was less than 50 per ,us = y&Q sin O/A, (7) cent. The low density powder represented by point e, where A is the area of the boundary, Substituting Fig. 7 may sinter more readily if grain growth removes for the curvature from the relation ps = -Ky&2 the the boundaries from some of the pores thus increasing surface curvature of porosity that neither shrinks nor the surface curvature to point g. These pores will now grows is negative and is given by the equation sinter by diflYnsionto other pores still intersected by K = --L sin S/A. A separate calculation of this boundaries or by diffusion to nearby boundaries. curvature shows that the pores always tend to siuter The curvature of the pores shrinking represented by when the dihedral angle 8 is 45’ 60’ I 75’ or 90”. point g, Pig. 7, descends the @ = 90 curve until their Indeed when the volume fraction of porosity is high size is equivalent to about 8 per cent porosity at and the surface curvature is negative there is a large which point the connecting links collapse isolating the driving force for sin&ring. This foroe results from the pore in the grain. This behavior has been observed surface tension forces acting on the small areas of in low density UO, powder compacts.(14) grain boundary. CONCLUSIONS The pores were also shown to be unstable with reThe model of grain edge porosity outlined is approspe~ttoneighboriugre~onsof slightly different volume fractions. Two pore volumes and their associate priate when discussing liquid preeipita~s, fission gas surfaoe curvature are represented by the points c and swelling and powder compacts. The morphology d, Fig. 7. The atom potential on the larger pore c is of the porosity is dependent on its contents only greater than that on the smaller pore d. Atoms will through the influence they have on the ratio ofsurface tend to migrate from c to d increasing the size of the to grain boundary energies. The model suooessfulIy predicts the shape of liquid precipitates having a larger pore at the expense of the smaller. dihedral angle of ~15’. Also the model demonstrates POWDER COMPACTS the long and short range stability of the precipitates. Interconnected porosity in powder compacts results Porosity with dihedral angles greater than 30” is from the formation of necks between adjacent shown to require a threshold volume before linkage particles. Initially the porosity may have no regular along the grain edges is locally stable. It is also shown
ACTA
138
however
that
this porosity
METALLURGICA,
is long range unstable,
a
large pore tending to grow at the expense of a smaller neighbor.
Finally
the
sintering
behavior
of UO,
powders may be explained by the changes in pore curvature associated with grain growth and change in green density. ACKNOWLEDGEMENT
This paper is published by permission of the Central Electricity
Generating
Board.
23,
1976
4. W. BEERE, Int. Conf. on Physical Elements. Berkelev. (19731.
Met. of Reactor
Fuel
anyi. A. ~L’TJRNBULL, ibid. 5. M. 0. T&ER 6. G. L. REYNOLDS and G. H. BANNISTER, J. Mat. Sci.
84 (1970).
5,
7. L. F. EPSTEIN and W. H. HOWLAND. J. Amer. Ceram. Sot. 36. 334 (19531. 185, 8. K. K. &ENY& and C. S. SMITH, Trans. A.I.M.E. 762 (1949). 9. 0. J. KLEPPA and J. A. WEIL, J. Am. Chem. Sot., 73, 4848 (1951). 176. 15 (1948). 10. C. S. SMITH. Trans. A.I.M.E. 11. G. L. REYNOLDS, W. B. BEERE and @. T.’ SAWBRIDOE, J. Nud. Mater. 41, 112 (1971). 12. H. L. FRISCH, J. M. HAMMERSLEY and D. J. A. WELSH,
Phye. Rev. 126, 949 (1962).
REFERENCES 1. C. S. SMITH, Met. Rev. 9, 1 (1964). 2. R. L. COBLE, J. A&. Phys. 32, 787 (1961). 3. W. B. BEERE and G. L. REYNOLDS, J. Nucl. 51 (1973).
VOL.
Mater.
47,
8, 1717 (1973). 14. W. BEERE,J. Mat.Sci. 15. W. D. KINGERY and B. FRAN(:OIS, Sintering and Related Phenomena, edited by G. C. KUCZYNSKI, p. 471. Gordon & Breach, New York (1967). 16. W. D. KINGERY and B. FRANCOIS, ibid. p. 499 (1967).