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The second stage sintering kinetics of powder compacts
THE
SECOND
STAGE
SINTERING
KINE’t’ICS
W.
OF
POWDER
COMPACTS*
BEEREt
A quantitative theory is presented of second stage powder sintering kinetics. The sintering rate is The porosity is assumed to have calculated from the stability of interconnected grain edge porosity. surfaces at equilibrium which maintain a constant contact angle with the grain boundaries. The sintering rate is shown to depend on the contact angle which in turn depends upon the ratio of surface to grain boundary energies. The sintering fluxes are shown to arise not only from excess vacancies at the porosity but also from the vacancy depletion on the grain boundaries. At high pore volume fractions the contribution to the fluxes from the grain boundary depletion greatly exceeds that from the excess vacancies at the pore surfaces. The second stage model presented is valid over a wide range of densities. The pore geometry considered may form immedately after the formation of necks between particles and the rounding of surfaces in the compact. The sintering rate of porosity with equilibrated surfaces is compared with sintering data for alumina powders. LA
CINETIQUE
DU
SECONDE
STADE
DU
FRITTAGE
D’UNE
POUDRE
COMTRIMEE
On pr&sente une thborie quantitative de la cin&ique du second stade du frittage d’une poudre. La vitesse de frittage est calculbe it partir de la stabilit6 de la porosit6 des bords de grains interconnect&. On suppose que les surfaces de la porosit6 sont en Bquilibre, et maintiennent un angle de contact constant avec les joints de grains. La vitesse de frittage depend de l’angle de co&act, qui depend B son tour du rapport de 1’6nergies de surface B 1’Bnergie de joints de grains. On montre que les flux de frittage proviennent non seulement des lacunes en exo&s Q la porosit6, mais Qgalement de 1’6puisement des lacunes aux joints de grains. Pour les taux volumiques de pores Blev&, la contribution de 1’Bpuisement aux joints de grains d&passe largement celle des lacunes en ex&s sur les surfaces de pores. Le mod&le du second stade present6 est valable pour un grand domaine de densit&. La g&om&trie des pores que l’on considere peut se former immbdiatement ap&s la formation de collets entre les particules et l’arrondissement des surfaces dan la matiBre compress&e. On compare la vitesse de frittage de la porosit6 avec des surfaces BquilibrBes avec des don&es sur le frittage des poudres d’alumine. ZWEITE
STUFE
SINTERKINETIK
VON
PULVERKOMPAUTMATERIALIEN
Eine quantitative Theorie der Kinetik der zweiten Stufe des Pulversinterns wird vorgelegt. Die Sintergeschwindigkeit wird aus der Stabilitkt der Sinterporositiit (grain edge porosity) berechnet. Es wird angenommen, da5 sich die Oberfliichen der Poren im Gleichgewicht befinden und mit den Korngrenzen einen konstanten Kontaktwinkel bilden. Die Sintergeschwindigkeit htingt vom Kontaktwinkel ab, der wiederum eine Funktion des Verhiiltnisses zwischen Oberfliichen- und Korngrenzenenergie ist. Der Sinterflu5 ist nicht nur eine Folge der ffberschu5leerstellen an den Poren, sondern such der Leerstellenverarmung an den Korngrenzen. Bei gro5en Porendichten iibersteigt der Anteil aus der Leerstellenverarmung an Korngrenzen den Anteil aus der Leerstelleniibersiittigung an den Porenoberfliichen bei weitem. Das vorgeschlagene Model1 gilt in einem gro5en Dichtebereich. Die betrachtete Porengeometrie kann direkt nach der Bildung van Htllsen zwischen Teilchen und dem Abrunden der Ober%chen entstehen. Die Sintergeschwindigkeit der Poren mit Oberf&chen im Gleichgewicht wird mit Sinterdaten von Aluminiumoxidpulver verglichen.
INTRODUCTION
The sintering
of powder compacts
into three stage@ physical mation
forms.
in which the porosity has separate The first stage
of bonds between
when the compact This
process
in rapid
Geometry of this type has been incorporated in second stage sintering models.(2*3) As the intercon-
may be divided
nected
deals with the for-
the particles
which form
is heated to elevated temperatures.
usually
sintering
takes
place
of several
quickly
per cent.
an
increasing
number
in the grains
grain
third
stage
boundaries.
The
of
or on
concerns
the
of the isolated pores.
The second
the
sinters
either
stability
resulting When
porosity
small pores are isolated
stage
extends
over a wider range
of
bonds between particles have grown to an appreciable
porosity than the 1st or 3rd stages and results in the largest increase in density. The second stage sintering
size relative
rate has been calculated
to the particle diameter
and the surfaces
have begun to smooth out the pore geometry enters a second stage. The porosity is situated around the grain edges meeting
at grain corners
throughout the compact. grain edges is intersected
and extending
The porosity along the by three grain boundaries.
pore surface area and the driving force for sintering. The driving force arose entirely from the cylindrical
Received April 22, 1974; revised July 8, 1974. t Berkeley Nuclear Laboratories, Berkeley, Gloucester, England. METALLURGICA,
VOL.
23,
JANUARY
1975
He constructed
length and the cylinder radius. The sintering rate of the porosity was taken to be proportional to the
*
ACTA
by Cable.(2)
a model consisting of cylindrical pores situated along the edges of tetrakaidecahedron grains. The volume of the porosity was calculated from the grain edge
pore curvature. 139
ACTA
140
~IETALLURGICA,
The assumptions implicit in Coble’s model appear reasonable at small pore volumes although the cylinders deviate in shape from those observed. A closer approach to observed pore geometry has recently been rnede(495) by calculating the shape of equilibThe pores were assumed rium grain edge porosity. to have complex curavature and to meet the grain boundaries at a constant dihedral angle satisfying the surface tension balance. The shapes found by Beeret4) were calculated by minimising the free energy of the pores by changing the pore shape at constant volume. Once the equilibrium shape was known the surface curvature, pore surface area and grain boundary area could be found for a given pore volume and dihedral angle. The values derived for the surface areas and curvatures enable an accurate assessment of the sintering rate of the equilibrium shapes. The present paper calculates these rates derived from the mod81(4) described below. The pore geometry is based on a corner unit of porosity which has tetrahedral symmetry, Fig. 1. The unit can be placed on a grain corner and the four projections connect with units on neighbouring grain corners. The method of linking the units is shown in Fig. 2 where 6 units are arranged to form an enclosed toroid of porosity. The units can be connected indefinitely forming extensive interlinked porosity. For the purposes of the model the units are distorted slightly and arranged round the edges of an idealised tetrakaideeahedron grain. The system of connected porosity is locally stable provided the volume is above a particular threshold volume. Below this volume the ~o~ections collapse and the porosity retreats to the grain corners. The
FIG. 1. A corner unit of porosity symmetry
and a grain boundary porosity.
having tetrahedral associated with the
VOL.
23,
1975
Fxa. 2. An arrangement
of six interconnecting corner units distorted slightly to fit on a hexagonal grain face. The pore shape is that predicted’*) for a dihedral angle of 75” and a pore volume fraction of 10 per cent.
process is analogous to a column of liquid breaking up as the diameter is reduced. The threshold volume depends on the dihedral angle 6 where 8 is given by 8 = COS-~(~~~2~~). When the dihedral angle is 90” the threshold volume is about 8 per cent porosity. Decreasing the dihedral angle decreases the threshold volume until when the dihedral angle is 30’ the porosity does not collapse even as the pore volume tends to zero.@) Increasing the volume of interconnected porosity decreases the area of grain boundary joining the grains. If the volume is increased to an upper threshold volume then the grain boundaries are consumed by the porosity. Above the upper threshold volume this particular type of in~rli~ing porosity cannot form. The upper threshold volume is about 67 per cent vol. fraction when the dihedral angle is 90” reducing to about 18 per cent when the dihedral angle is 30”. Thus the second stage sintering model covers the range from the upper limit, when the type of porosity can form, down to the lower limit when the connections break up into individual pores. For a powder compact having a dihedral angle of 75” the range extends from about 55 per cent to about 6 per cent pore fractions. If the powder is compressed to a green density of 50 per cent then the pore geometry is expected to enter the second stage immedia~ly after the formation of necks and the rounding of surfaces. Although the grain edge porosity envisaged can form locally stable connections between grain edges, the porosity is long-range unstable. If for instance two neighbouring pores have small differences in size the curvature on the larger pore is less than on the smaller pore. Curvature is defined in the sense
BEERE:
SINTERING
KINETICS
that a spherical cavity has positive surface curvature. The difference in curvature results in a difference in vacancy potential just below the two surfaces. The resulting vacancy flux causes the larger cavity to grow at the expense of the smaller. The process is analogous to Ostwald ripening.(7) When however the dihedral angle is reduced to 30” or less the opposite is the case and the small pore grows at the expense of the larger evening out any differences in size. Such small dihedral angles are not met in powder compacts but are usual for liquid phases in polycrystals. The flux of atoms across the pore surface does not of course increase the density of the compact but may effect the stability of some pores-thus influencing the sintering kinetics.(s) VOLUME
COMPACTS
141
25 96
---______C_O_BLE
IL
0
_J
O-I 0.2 O-3 OL 0 5 0 6 0 7
PORE
VOLUME
FRACTION,
AVIV
sin&ringforce versus pore volume fraction for dihedral angles from 15’ to 90’.
The shrinkage rate of the interconnected porosity depends on the driving force for sintering. Excess vacancies created on the pore surface migrate randomly through the grain to a grain boundary. The excess vacancy concentration immediately under the pore surface is given by C, = C, exp (Ky,C+‘cT), where C, is the equilibrium concentration over a plane surface, K is the surface curvature, ys the surface tension, Q the atomic volume, k Boltzmann’s constant and T is absolute temperature. When vacancies annihilate on the boundary the grains move together and work is done on the pore surfaces. The change is Gibbs free energy per vacancy AG is (1)
where AH enthalpy of vacancy formation As the change in entropy, C, the vacancy concentration at the boundary and L and A are the circumferential length and area of boundary situated on a grain face, Fig. 1. A grain boundary acting as a perfect sink maintains an equilibrium concentration of vacancies and the change in Gibbs free energy is zero. From equation (1) the vacancy concentration c is C, = C, exp (-Ly,Q
POWDER
FIG. 3. The
DIFFUSION
AG=AH-TAS+kTlnC,+Ly,F,
OF
sin B/AkT)
The driving force for sintering is given by the difference in vacancy concentration. Usually KysQ < kT and the driving force is given by :
equilibrium interconnected porosity. The product is independent of grain size and is plotted versus volume fraction porosity in Fig. 3 for dihedral angles ranging from 15” to 90”. When the dihedral angle is equal to or greater than 45” decreasing the pore volume increases the surface curvature K and hence the sintering force. Increasing the pore volume decreases the grain boundary area A, increasing the term L sin 8/A and also the sintering force. At small pore fractions the surface curvature term K predominates whilst at large pore fractions the grain boundary term L sin B/A is dominant. At intermediate volumes the sintering force passes through a minimum. The sintering force arising solely from the surface curvature of cylinders arranged along the grain edges@) is also shown in Fig. 3. At the lower threshold volume the sintering force is in approximate agreement. At large pore volumes the difference arises mainly from the inclusion of the grain boundary vacancy concentration term L sin O/A in the present work. Below 30 per cent porosity decreasing the dihedral angle 19 decreases the sintering force. When the dihedral angle is 15” the sintering force is negative indicating that the pores swell. This agrees with the observed behaviour of liquid precipitates(4) The flux of vacancies leaving the pore and entering the boundary is given by :
j = -o,,(aclax), The product of the bracketed term in equation (2) and the grain edge length 1 was computed for the
(3)
where D,, is the vacancy diffusion coefficient and &lax is the vacancy concentration gradient. The
142
ACTA
METALLURGICA,
rate of change of volume of a corner unit is given by :
VOL.
volume
23,
1975
fraction
6Bu/81/2/3,
S(AV/V) is given by S(AV/V) = where 82/213 is the volume of a tetrakai-
decahedron
and 6u is the change in volume
to the sintering where A is the area of boundary unit.
The
boundary
or pore per corner
area per corner
unit is more
often smaller than the surface area of a corner unit, The value of A was put equal to the smaller
Fig. 4.
of the two areas. &/ax
Putting
gradient
then the rate of change of volume
of a corner
unit is :
where D, is the volume self diffusion by D, = D,,C&. by equation vacancies the
leaving
the grains move still further.
together
at high
rate of change
multiplying
reducing
equation
is the area of grain boundary ence of porosity. be expressed are 24
The volume
as a volume
corner
units
per
by
Since however boundaries,
the pore volume volumes.
the (1 + A,/A) fraction is :
dW’/V) _
term,
=so,
the rate
of change
of pore
K
8 J”kT
(6) During sintering the cube of the grain size of many compacts increases linearly with time. The grain edge length 1 is a constant diameter.
Putting
2.1212, where 1, is the initial temperature
fraction
l3 = 1,3 + bt
dependent
and
of the grain A + A, =
value of 1 and 6 is a
constant,,
equation
(6) be-
comes
The
is obtained
(1 + A,/A) removed
where
by A,
by the pres-
of a corner unit can
fraction
grain
and so does not
to the volume
pore
of volume
(5) by
described
removed
on the grain
The latter contribution
is substantial
increased
coefficient given
the pore surface.
annihilate
’
in volume
(5) is due only to volume
vacancies
change
The change
1
(5)
sweeping
due only
that
contribute to the change in pore fraction. Substituting A V/ V for u in equation (5) and incorporating
dt By, =
It is assumed
results in porosity
C,)B/l, where B is a geometrical
equal to (C, -
factor,
the concentration
growth
fluxes.
porosity.
There
tetrakaidecahedron
grain
each shared by four grains, thus the change in pore
(7) Where
(AV/V)
is the volume
time t and (AV/V), of second
stage sintering.
is proportional
grain size squared
to grain size.
(7) is dimensionless
represented
curvature
Consequently
If the integral by I, then
K
of grain size for a
Also areas are proportional
equation
at a
at the onset
and grain edge lengths
portional grain size.
porosity
The surface
to the reciprocal
given pore fraction.
4r
fraction
is the pore volume
and
to
are pro-
each side of
independent
of
on the left hand side is
(8)
??? \
The value of I,
4 %
was found
by numerical
integra-
tion for dihedral angles of 45, 60, 75 and 90”, Fig. 5, was the maximum pore volume where (AV/V), allowed by the model. The relationship between I, and AV/ V can be represented accurately by a quadratic equation. The equations corresponding to the four dihedral angles, Fig. 5, are listed in Table 1. 01
02
03
0.4
0.5
PORE VOLUME FRACTION
06
07
AL’/ V
FIG. 4. The grain boundary and surface area per corner unit versus pore volume fraction for dihedral angles of 45” and 90”.
If the curves of Fig. 5 are represented by a straight line then a simpler equation is obtained at the expense of up to a factor of 2 error. When the dihedral angle is 75’ then the slope, Fig. 5, is about -0.145 If the second stage commences at a pore fraction
BEERE:
*08
.
.
.l
.2
SINTERING
.
.
.
.
.5
.6
KINETICS
OF
POWDER
COMPACTS
143
.07 .06 .05 d .01 .03 .02 .Ol O0
.3 4
_ 7
AV/ V
AV/ V
Fra. 5. The value of the integral 1, for volume diffusion.
FIG. 6. The value of the integral 1, for grain boundary
diffusion.
TABLE
1. IL = a + b(y)
TABLET.
&=a+b(y)
c
Dihedral angle
a
0.077 0.161 0.100 0.252
90’ 75” 60” 45”
+ c(y)’
+c(y)
.-Dihedral angle
a
90”
-0.187 -0.247 -0.240 -0.349
0.089
75”
(AV/V),
6
0.086 0.080 0.082
then the pore fraction
at a time t is given by
integrated
s
0.0264 0.0240 0.0240 0.0273
-
(AV/Vh,
SJz.145 GRAIN
BOUNDARY
The shrinkage mechanism
-0.0891 - 0.0995 -0.116 -0.178
0.0747 0.102 0.140 0.290
(A
+
WJ’P’) A,)
q!$q;-;),
diffusion
(11)
depends on the same driving forces as the
volume diffusion mechanism. of vacancy
A
DIFFUSION
rate for a grain boundary
c
to give
(AVIV)
12
6
path produces
The different geometry
a change in rate constant.
where P = la3 + bt. The integral I, on the left hand side of equation
(11) was integrated
The shrinkage rate has been calculated in the Appendix
the values are plotted
assuming
of 45’, 60”, 75” and 90”.
that
vacancies
originating
diffuse in the plane of the boundary
at
the
pore
and plate out
sented
accurately
by
numerically
in Fig. 6 for dihedral The integral the quadratic
and
angles
can be reprerelationships
evenly over the whole area of boundary. Diffusion on the pore surface is assumed to maintain an equi-
given in Table 2. The integral may be represented
librium pore shape replenishing tering the boundary. The rate
if an error of up to a factor of ~5 is acceptable. When the dihedral angle is 75” the average slope from
volume fraction
the vacancies enof change of pore
Fig. 6 is -0.04
with time is
and the porosity
WV/V --__!z(?.gy(
by a straight line
at time t is given by
(12)
dt where 13 = PO + bt. where 6 is the grain boundary
width.
If it is again
assumed that the cube of the grain edge length 1 increases linearly with time then equation 10 can be
DISCUSSION
The sintering rate predicted by equation (9) is essentially similar to Cable’s@) equation for the
ACTA
144
METALLURCICA,
sintering rate of cylinders on grain edges. The predicted sintering rates are equal if the geomet~oal constant B in equation (9) is put equal to 1.4. In practice B is likely to be about a factor of three less. Closer agreement with Coble’s theory may be obtained by comparison with the exact expression at the lower threshold volume. Better agreement is expected at small pore fractions because the assumptions made in the derivations are then the same. These are that the sintering force is proportional to the curvature of the porosity and the flux of vacancies is proportional to the area of the porosity. At large pore fractions these assumptions no longer hold true. The pore curvature decreases but the six&ring force increases owing to the rapidly decreasing vacancy concentration on the grain boundaries. Also the area of grain boundary is much smaller than the pore area and consequently the total vacancy flux will depend more critically on grain boundary area. For these reasons the agreement at low pore volume fractions is not maintained at high porositics. The sintering behaviour of alumina powder annealed at 1550% is plotted according to equation (8) in Fig. 7. The sintering data,(s) gives the variation of density and grain size I) with time. The open pore volume was calculated by assuming all the porosity to be open. This assumption is only valid during the early stages of sintering. The value of the integral I, was obtained from Figs. 5 and 7, was plotted assuming the grain size at the onset of second stage sintering to be 0.3 pm. Equation (8) predicts that the ex~rimental data should pass through the origin. The data can be forced through the origin by changing the disposable parameters which are the grain size D, and the pore volume fraction ~AViV~~ at the onset X)7/” XI6. ,05. ,04 @.-ii,03X)2”
c
Fra 7. Alumina sinking dete(*J plotted according to a volume diffusion mechanism, (a) (ATI/V), = 0.55, Do = 0.3 ,um, (b) (AT’/v), = 0.45, D, = 0.6 ,um, (c) (AV/V), = 0.55, D, = 0.6pm.
VOL.
23,
1975
of the second stage. An alteration to these moves the experimental points as a group sideways or up and down but does not alter the slope. The slope is equal to 12DLBy,~;1/82/2 kTb. Taking the steepest gradient at the onset of sintering and putting B = 0.5, y = 1 J/ma, Q = 6 x 1O-29M2, k = 1.38 x lo-z3 J/K, T = 1823 K and b = 3.8 x 1O-21majsec (from Ref. (8) then the volume self diffusion coef%oient is 5 x 10-r* mzfsee. This value is up to about an order of magnitude higher than values calculated from the diffusion creep of alumina(B) or from initial sintering rates.(lO) The curvature of Fig. 7 is reduced if account is taken of closed porosity contributing to the overall pore fraction. If 4 per cent of the porosity is closed when the total porosity is 12 per cent then the plot in Fig. 7 becomes a straight line. It is important to consider only open interconnected porosity because closed pores will sinter at a different rate. If the closed pores contain gas the shrinkage rate may be very slow. CONCLUSEONS
The sinteing rate of equilibrium grain edge porosity may usefully be applied to the second stage sir&ring kinetics of powders. The model is valid between an upper and a lower threshold volume. Many powder compacts are expected to be below the upper threshold volume immediately after the formation of necks and the rounding of surfaces, Below the lower threshold volume the interconnected structure is unstable and porosity forms isolated pores. The sintering rate depends not only on the surface curvature but also on the grain boundary vacancy potential. At large pore volumes the boundary areas are relatively small and the vacancy oon~entration is greatly suppressed. As the pore volume decreases the boundary areas grow, the vacancy concentration increases and so the sintering force decreases, Simultaneously the pore surface curvature increases with decreasing pore volume. The increased curvature increases the local vacancy concentr&tion and at small pore volumes the sintering force increases. At intermediate pore volumes the sintering force passes through a minimum. During most of the second stage the grain boundary area is much smaller than the pore surface area. Co~equently the sintering rate is more sensitive to grain boundary area than pore area. The dihedral angle maintained between pore surface and grain boundary also affects sintering rate. If the angle is reduced to values observed in liquid precipitates the volume of precipitate grows rather than sinters.
BEERE:
The
quantitative
sintering
calculation
rates allows
the assessment data
SINTERING
a more
of
for straight
external
density.
atmosphere
Failure
When
coefficients relaxation
and
pore
to
interpreting
OF
where
POWDER
with
is relevant are
* = 0 dr
when
P(R) = Kay,
(15) R
27rRy, sin 8 = Equation
-
g
T/J(T) dr.
s0
(14) ensures that the vacancy
centre of the boundary
is zero.
the potential
on the pore surface.
may
rium concentration
many
ceramics
measurements
have
for
diffusion
creep
interfacial
kinetics may be rate controlling.
which
shown
that
is annihilated
equal to
If the boundary
maintains
of vacancies.
(15) fixes
an equilib-
When
a vacancy
the change in Gibbs free energy is zero.
Equation (16) meets this condition by balancing the change in chemical potential over the boundary
ACKNOWLEDGEMENT
This paper is published by permission Electricity Generating Board.
is a perfect sink the boundary
(16)
flux at the
Equation
It is also necessary for the grain boundaries to act as perfect sinks for vacancies. The last requirement for
of the central
with the work done on the external
restraint.
chemical potential
is
over the boundary
The
(17)
REFERENCES 1. J. WHITE, Sintering and Related Phenowaena, edited by KUCZYNSKI, p. 354. Gordon & Breach, New York (1967). 2. R. L. COBLE, J. uppl. Phys. 32, 787 (1961). 3. W. BEERE, J. Mat. Sci. 8, 1717 (1973). 4. W. BEERE, to be published. 5. M. 0. TUCKER and J. A. TURNBULL, Int. Conf. on Physical Met. of Reactor Fuel Ekments, Berkeley, England (1973). 6. C. S. SMITE, Met. Rev. 9, 1 (1964). W. OSTWALD, 2. Phys. Chena. 84, 495 (1900). :: R. L. COBLE.J. a~&. Phvs. 32. 793 (1961). 9. C. W. HEW&N c&d’ W. 0: K&&RY,‘ J. Ana. Ceram. Sot. 50, 218 (1967). 10. W. R. RAO and I. B. CUTLER, J. Am. Ceram. Sot. 55, 170 (1972). 11. B. BURTON and G. L. REYNOLDS, Acta Met. 21. 1073 (1973). APPENDIX
The sintering vacancy
rate is calculated
flux entering
by grain edge porosity.
where cc is a constant.
The flux at the pore surface
a grain boundary The boundary
deriving
the
and the total flux is J = j2rrRd. change of volume
Hence the rate of
(19) A tetrakaidecahedron by two grains. tion
porosity
has
14 faces
equation
is assumed
fraction
vacancy potential at a distance r from the boundary centre, then the potential satisfies the differential
each
A small change in the volume
AV/V
is equal
for a circle is equal to 2/R.
surrounded If p(r) is the
is
(18)
to
shared frac-
6(AV/ V) = 7 SU/
SXJ’% and the ratio of circumference by
to be circular, Fig. 1, and of radius R.
to area, L/A,
Incorporating
these into
(19) gives the rate of change of pore volume
dW/V -=-~(!%$j(~+K)(A+AA’)_ dt
equation
(20) The last term in brackets (13)
10
of
(14)
on the rim of the boundary
true
rate
r = 0
the potential
hold
to volumetric
surface
the
(13) is solved
changes.
not
is rapid compared
diffusion
that
on
Equation
subject to the conditions
increasing
when
such
depending
of vacancies.
it is
for this causes difficulty
sizes
145
COMPACTS
/l is a constant
annihilation
porosity. During not connected to
increases
to allow
in correct interpretation. The theory presented
stage
approach
line behaviour
important to consider only open sintering the fraction of porosity the
second
rigorous
of mechanisms.
and looking
KINETICS
(20) accounts
on the right of equation
for the reduction
the grains move together.
in pore volume
as
SECOND
STAGE
SINTERING
KINE’t’ICS
W.
OF
POWDER
COMPACTS*
BEEREt
A quantitative theory is presented of second stage powder sintering kinetics. The sintering rate is The porosity is assumed to have calculated from the stability of interconnected grain edge porosity. surfaces at equilibrium which maintain a constant contact angle with the grain boundaries. The sintering rate is shown to depend on the contact angle which in turn depends upon the ratio of surface to grain boundary energies. The sintering fluxes are shown to arise not only from excess vacancies at the porosity but also from the vacancy depletion on the grain boundaries. At high pore volume fractions the contribution to the fluxes from the grain boundary depletion greatly exceeds that from the excess vacancies at the pore surfaces. The second stage model presented is valid over a wide range of densities. The pore geometry considered may form immedately after the formation of necks between particles and the rounding of surfaces in the compact. The sintering rate of porosity with equilibrated surfaces is compared with sintering data for alumina powders. LA
CINETIQUE
DU
SECONDE
STADE
DU
FRITTAGE
D’UNE
POUDRE
COMTRIMEE
On pr&sente une thborie quantitative de la cin&ique du second stade du frittage d’une poudre. La vitesse de frittage est calculbe it partir de la stabilit6 de la porosit6 des bords de grains interconnect&. On suppose que les surfaces de la porosit6 sont en Bquilibre, et maintiennent un angle de contact constant avec les joints de grains. La vitesse de frittage depend de l’angle de co&act, qui depend B son tour du rapport de 1’6nergies de surface B 1’Bnergie de joints de grains. On montre que les flux de frittage proviennent non seulement des lacunes en exo&s Q la porosit6, mais Qgalement de 1’6puisement des lacunes aux joints de grains. Pour les taux volumiques de pores Blev&, la contribution de 1’Bpuisement aux joints de grains d&passe largement celle des lacunes en ex&s sur les surfaces de pores. Le mod&le du second stade present6 est valable pour un grand domaine de densit&. La g&om&trie des pores que l’on considere peut se former immbdiatement ap&s la formation de collets entre les particules et l’arrondissement des surfaces dan la matiBre compress&e. On compare la vitesse de frittage de la porosit6 avec des surfaces BquilibrBes avec des don&es sur le frittage des poudres d’alumine. ZWEITE
STUFE
SINTERKINETIK
VON
PULVERKOMPAUTMATERIALIEN
Eine quantitative Theorie der Kinetik der zweiten Stufe des Pulversinterns wird vorgelegt. Die Sintergeschwindigkeit wird aus der Stabilitkt der Sinterporositiit (grain edge porosity) berechnet. Es wird angenommen, da5 sich die Oberfliichen der Poren im Gleichgewicht befinden und mit den Korngrenzen einen konstanten Kontaktwinkel bilden. Die Sintergeschwindigkeit htingt vom Kontaktwinkel ab, der wiederum eine Funktion des Verhiiltnisses zwischen Oberfliichen- und Korngrenzenenergie ist. Der Sinterflu5 ist nicht nur eine Folge der ffberschu5leerstellen an den Poren, sondern such der Leerstellenverarmung an den Korngrenzen. Bei gro5en Porendichten iibersteigt der Anteil aus der Leerstellenverarmung an Korngrenzen den Anteil aus der Leerstelleniibersiittigung an den Porenoberfliichen bei weitem. Das vorgeschlagene Model1 gilt in einem gro5en Dichtebereich. Die betrachtete Porengeometrie kann direkt nach der Bildung van Htllsen zwischen Teilchen und dem Abrunden der Ober%chen entstehen. Die Sintergeschwindigkeit der Poren mit Oberf&chen im Gleichgewicht wird mit Sinterdaten von Aluminiumoxidpulver verglichen.
INTRODUCTION
The sintering
of powder compacts
into three stage@ physical mation
forms.
in which the porosity has separate The first stage
of bonds between
when the compact This
process
in rapid
Geometry of this type has been incorporated in second stage sintering models.(2*3) As the intercon-
may be divided
nected
deals with the for-
the particles
which form
is heated to elevated temperatures.
usually
sintering
takes
place
of several
quickly
per cent.
an
increasing
number
in the grains
grain
third
stage
boundaries.
The
of
or on
concerns
the
of the isolated pores.
The second
the
sinters
either
stability
resulting When
porosity
small pores are isolated
stage
extends
over a wider range
of
bonds between particles have grown to an appreciable
porosity than the 1st or 3rd stages and results in the largest increase in density. The second stage sintering
size relative
rate has been calculated
to the particle diameter
and the surfaces
have begun to smooth out the pore geometry enters a second stage. The porosity is situated around the grain edges meeting
at grain corners
throughout the compact. grain edges is intersected
and extending
The porosity along the by three grain boundaries.
pore surface area and the driving force for sintering. The driving force arose entirely from the cylindrical
Received April 22, 1974; revised July 8, 1974. t Berkeley Nuclear Laboratories, Berkeley, Gloucester, England. METALLURGICA,
VOL.
23,
JANUARY
1975
He constructed
length and the cylinder radius. The sintering rate of the porosity was taken to be proportional to the
*
ACTA
by Cable.(2)
a model consisting of cylindrical pores situated along the edges of tetrakaidecahedron grains. The volume of the porosity was calculated from the grain edge
pore curvature. 139
ACTA
140
~IETALLURGICA,
The assumptions implicit in Coble’s model appear reasonable at small pore volumes although the cylinders deviate in shape from those observed. A closer approach to observed pore geometry has recently been rnede(495) by calculating the shape of equilibThe pores were assumed rium grain edge porosity. to have complex curavature and to meet the grain boundaries at a constant dihedral angle satisfying the surface tension balance. The shapes found by Beeret4) were calculated by minimising the free energy of the pores by changing the pore shape at constant volume. Once the equilibrium shape was known the surface curvature, pore surface area and grain boundary area could be found for a given pore volume and dihedral angle. The values derived for the surface areas and curvatures enable an accurate assessment of the sintering rate of the equilibrium shapes. The present paper calculates these rates derived from the mod81(4) described below. The pore geometry is based on a corner unit of porosity which has tetrahedral symmetry, Fig. 1. The unit can be placed on a grain corner and the four projections connect with units on neighbouring grain corners. The method of linking the units is shown in Fig. 2 where 6 units are arranged to form an enclosed toroid of porosity. The units can be connected indefinitely forming extensive interlinked porosity. For the purposes of the model the units are distorted slightly and arranged round the edges of an idealised tetrakaideeahedron grain. The system of connected porosity is locally stable provided the volume is above a particular threshold volume. Below this volume the ~o~ections collapse and the porosity retreats to the grain corners. The
FIG. 1. A corner unit of porosity symmetry
and a grain boundary porosity.
having tetrahedral associated with the
VOL.
23,
1975
Fxa. 2. An arrangement
of six interconnecting corner units distorted slightly to fit on a hexagonal grain face. The pore shape is that predicted’*) for a dihedral angle of 75” and a pore volume fraction of 10 per cent.
process is analogous to a column of liquid breaking up as the diameter is reduced. The threshold volume depends on the dihedral angle 6 where 8 is given by 8 = COS-~(~~~2~~). When the dihedral angle is 90” the threshold volume is about 8 per cent porosity. Decreasing the dihedral angle decreases the threshold volume until when the dihedral angle is 30’ the porosity does not collapse even as the pore volume tends to zero.@) Increasing the volume of interconnected porosity decreases the area of grain boundary joining the grains. If the volume is increased to an upper threshold volume then the grain boundaries are consumed by the porosity. Above the upper threshold volume this particular type of in~rli~ing porosity cannot form. The upper threshold volume is about 67 per cent vol. fraction when the dihedral angle is 90” reducing to about 18 per cent when the dihedral angle is 30”. Thus the second stage sintering model covers the range from the upper limit, when the type of porosity can form, down to the lower limit when the connections break up into individual pores. For a powder compact having a dihedral angle of 75” the range extends from about 55 per cent to about 6 per cent pore fractions. If the powder is compressed to a green density of 50 per cent then the pore geometry is expected to enter the second stage immedia~ly after the formation of necks and the rounding of surfaces. Although the grain edge porosity envisaged can form locally stable connections between grain edges, the porosity is long-range unstable. If for instance two neighbouring pores have small differences in size the curvature on the larger pore is less than on the smaller pore. Curvature is defined in the sense
BEERE:
SINTERING
KINETICS
that a spherical cavity has positive surface curvature. The difference in curvature results in a difference in vacancy potential just below the two surfaces. The resulting vacancy flux causes the larger cavity to grow at the expense of the smaller. The process is analogous to Ostwald ripening.(7) When however the dihedral angle is reduced to 30” or less the opposite is the case and the small pore grows at the expense of the larger evening out any differences in size. Such small dihedral angles are not met in powder compacts but are usual for liquid phases in polycrystals. The flux of atoms across the pore surface does not of course increase the density of the compact but may effect the stability of some pores-thus influencing the sintering kinetics.(s) VOLUME
COMPACTS
141
25 96
---______C_O_BLE
IL
0
_J
O-I 0.2 O-3 OL 0 5 0 6 0 7
PORE
VOLUME
FRACTION,
AVIV
sin&ringforce versus pore volume fraction for dihedral angles from 15’ to 90’.
The shrinkage rate of the interconnected porosity depends on the driving force for sintering. Excess vacancies created on the pore surface migrate randomly through the grain to a grain boundary. The excess vacancy concentration immediately under the pore surface is given by C, = C, exp (Ky,C+‘cT), where C, is the equilibrium concentration over a plane surface, K is the surface curvature, ys the surface tension, Q the atomic volume, k Boltzmann’s constant and T is absolute temperature. When vacancies annihilate on the boundary the grains move together and work is done on the pore surfaces. The change is Gibbs free energy per vacancy AG is (1)
where AH enthalpy of vacancy formation As the change in entropy, C, the vacancy concentration at the boundary and L and A are the circumferential length and area of boundary situated on a grain face, Fig. 1. A grain boundary acting as a perfect sink maintains an equilibrium concentration of vacancies and the change in Gibbs free energy is zero. From equation (1) the vacancy concentration c is C, = C, exp (-Ly,Q
POWDER
FIG. 3. The
DIFFUSION
AG=AH-TAS+kTlnC,+Ly,F,
OF
sin B/AkT)
The driving force for sintering is given by the difference in vacancy concentration. Usually KysQ < kT and the driving force is given by :
equilibrium interconnected porosity. The product is independent of grain size and is plotted versus volume fraction porosity in Fig. 3 for dihedral angles ranging from 15” to 90”. When the dihedral angle is equal to or greater than 45” decreasing the pore volume increases the surface curvature K and hence the sintering force. Increasing the pore volume decreases the grain boundary area A, increasing the term L sin 8/A and also the sintering force. At small pore fractions the surface curvature term K predominates whilst at large pore fractions the grain boundary term L sin B/A is dominant. At intermediate volumes the sintering force passes through a minimum. The sintering force arising solely from the surface curvature of cylinders arranged along the grain edges@) is also shown in Fig. 3. At the lower threshold volume the sintering force is in approximate agreement. At large pore volumes the difference arises mainly from the inclusion of the grain boundary vacancy concentration term L sin O/A in the present work. Below 30 per cent porosity decreasing the dihedral angle 19 decreases the sintering force. When the dihedral angle is 15” the sintering force is negative indicating that the pores swell. This agrees with the observed behaviour of liquid precipitates(4) The flux of vacancies leaving the pore and entering the boundary is given by :
j = -o,,(aclax), The product of the bracketed term in equation (2) and the grain edge length 1 was computed for the
(3)
where D,, is the vacancy diffusion coefficient and &lax is the vacancy concentration gradient. The
142
ACTA
METALLURGICA,
rate of change of volume of a corner unit is given by :
VOL.
volume
23,
1975
fraction
6Bu/81/2/3,
S(AV/V) is given by S(AV/V) = where 82/213 is the volume of a tetrakai-
decahedron
and 6u is the change in volume
to the sintering where A is the area of boundary unit.
The
boundary
or pore per corner
area per corner
unit is more
often smaller than the surface area of a corner unit, The value of A was put equal to the smaller
Fig. 4.
of the two areas. &/ax
Putting
gradient
then the rate of change of volume
of a corner
unit is :
where D, is the volume self diffusion by D, = D,,C&. by equation vacancies the
leaving
the grains move still further.
together
at high
rate of change
multiplying
reducing
equation
is the area of grain boundary ence of porosity. be expressed are 24
The volume
as a volume
corner
units
per
by
Since however boundaries,
the pore volume volumes.
the (1 + A,/A) fraction is :
dW’/V) _
term,
=so,
the rate
of change
of pore
K
8 J”kT
(6) During sintering the cube of the grain size of many compacts increases linearly with time. The grain edge length 1 is a constant diameter.
Putting
2.1212, where 1, is the initial temperature
fraction
l3 = 1,3 + bt
dependent
and
of the grain A + A, =
value of 1 and 6 is a
constant,,
equation
(6) be-
comes
The
is obtained
(1 + A,/A) removed
where
by A,
by the pres-
of a corner unit can
fraction
grain
and so does not
to the volume
pore
of volume
(5) by
described
removed
on the grain
The latter contribution
is substantial
increased
coefficient given
the pore surface.
annihilate
’
in volume
(5) is due only to volume
vacancies
change
The change
1
(5)
sweeping
due only
that
contribute to the change in pore fraction. Substituting A V/ V for u in equation (5) and incorporating
dt By, =
It is assumed
results in porosity
C,)B/l, where B is a geometrical
equal to (C, -
factor,
the concentration
growth
fluxes.
porosity.
There
tetrakaidecahedron
grain
each shared by four grains, thus the change in pore
(7) Where
(AV/V)
is the volume
time t and (AV/V), of second
stage sintering.
is proportional
grain size squared
to grain size.
(7) is dimensionless
represented
curvature
Consequently
If the integral by I, then
K
of grain size for a
Also areas are proportional
equation
at a
at the onset
and grain edge lengths
portional grain size.
porosity
The surface
to the reciprocal
given pore fraction.
4r
fraction
is the pore volume
and
to
are pro-
each side of
independent
of
on the left hand side is
(8)
??? \
The value of I,
4 %
was found
by numerical
integra-
tion for dihedral angles of 45, 60, 75 and 90”, Fig. 5, was the maximum pore volume where (AV/V), allowed by the model. The relationship between I, and AV/ V can be represented accurately by a quadratic equation. The equations corresponding to the four dihedral angles, Fig. 5, are listed in Table 1. 01
02
03
0.4
0.5
PORE VOLUME FRACTION
06
07
AL’/ V
FIG. 4. The grain boundary and surface area per corner unit versus pore volume fraction for dihedral angles of 45” and 90”.
If the curves of Fig. 5 are represented by a straight line then a simpler equation is obtained at the expense of up to a factor of 2 error. When the dihedral angle is 75’ then the slope, Fig. 5, is about -0.145 If the second stage commences at a pore fraction
BEERE:
*08
.
.
.l
.2
SINTERING
.
.
.
.
.5
.6
KINETICS
OF
POWDER
COMPACTS
143
.07 .06 .05 d .01 .03 .02 .Ol O0
.3 4
_ 7
AV/ V
AV/ V
Fra. 5. The value of the integral 1, for volume diffusion.
FIG. 6. The value of the integral 1, for grain boundary
diffusion.
TABLE
1. IL = a + b(y)
TABLET.
&=a+b(y)
c
Dihedral angle
a
0.077 0.161 0.100 0.252
90’ 75” 60” 45”
+ c(y)’
+c(y)
.-Dihedral angle
a
90”
-0.187 -0.247 -0.240 -0.349
0.089
75”
(AV/V),
6
0.086 0.080 0.082
then the pore fraction
at a time t is given by
integrated
s
0.0264 0.0240 0.0240 0.0273
-
(AV/Vh,
SJz.145 GRAIN
BOUNDARY
The shrinkage mechanism
-0.0891 - 0.0995 -0.116 -0.178
0.0747 0.102 0.140 0.290
(A
+
WJ’P’) A,)
q!$q;-;),
diffusion
(11)
depends on the same driving forces as the
volume diffusion mechanism. of vacancy
A
DIFFUSION
rate for a grain boundary
c
to give
(AVIV)
12
6
path produces
The different geometry
a change in rate constant.
where P = la3 + bt. The integral I, on the left hand side of equation
(11) was integrated
The shrinkage rate has been calculated in the Appendix
the values are plotted
assuming
of 45’, 60”, 75” and 90”.
that
vacancies
originating
diffuse in the plane of the boundary
at
the
pore
and plate out
sented
accurately
by
numerically
in Fig. 6 for dihedral The integral the quadratic
and
angles
can be reprerelationships
evenly over the whole area of boundary. Diffusion on the pore surface is assumed to maintain an equi-
given in Table 2. The integral may be represented
librium pore shape replenishing tering the boundary. The rate
if an error of up to a factor of ~5 is acceptable. When the dihedral angle is 75” the average slope from
volume fraction
the vacancies enof change of pore
Fig. 6 is -0.04
with time is
and the porosity
WV/V --__!z(?.gy(
by a straight line
at time t is given by
(12)
dt where 13 = PO + bt. where 6 is the grain boundary
width.
If it is again
assumed that the cube of the grain edge length 1 increases linearly with time then equation 10 can be
DISCUSSION
The sintering rate predicted by equation (9) is essentially similar to Cable’s@) equation for the
ACTA
144
METALLURCICA,
sintering rate of cylinders on grain edges. The predicted sintering rates are equal if the geomet~oal constant B in equation (9) is put equal to 1.4. In practice B is likely to be about a factor of three less. Closer agreement with Coble’s theory may be obtained by comparison with the exact expression at the lower threshold volume. Better agreement is expected at small pore fractions because the assumptions made in the derivations are then the same. These are that the sintering force is proportional to the curvature of the porosity and the flux of vacancies is proportional to the area of the porosity. At large pore fractions these assumptions no longer hold true. The pore curvature decreases but the six&ring force increases owing to the rapidly decreasing vacancy concentration on the grain boundaries. Also the area of grain boundary is much smaller than the pore area and consequently the total vacancy flux will depend more critically on grain boundary area. For these reasons the agreement at low pore volume fractions is not maintained at high porositics. The sintering behaviour of alumina powder annealed at 1550% is plotted according to equation (8) in Fig. 7. The sintering data,(s) gives the variation of density and grain size I) with time. The open pore volume was calculated by assuming all the porosity to be open. This assumption is only valid during the early stages of sintering. The value of the integral I, was obtained from Figs. 5 and 7, was plotted assuming the grain size at the onset of second stage sintering to be 0.3 pm. Equation (8) predicts that the ex~rimental data should pass through the origin. The data can be forced through the origin by changing the disposable parameters which are the grain size D, and the pore volume fraction ~AViV~~ at the onset X)7/” XI6. ,05. ,04 @.-ii,03X)2”
c
Fra 7. Alumina sinking dete(*J plotted according to a volume diffusion mechanism, (a) (ATI/V), = 0.55, Do = 0.3 ,um, (b) (AT’/v), = 0.45, D, = 0.6 ,um, (c) (AV/V), = 0.55, D, = 0.6pm.
VOL.
23,
1975
of the second stage. An alteration to these moves the experimental points as a group sideways or up and down but does not alter the slope. The slope is equal to 12DLBy,~;1/82/2 kTb. Taking the steepest gradient at the onset of sintering and putting B = 0.5, y = 1 J/ma, Q = 6 x 1O-29M2, k = 1.38 x lo-z3 J/K, T = 1823 K and b = 3.8 x 1O-21majsec (from Ref. (8) then the volume self diffusion coef%oient is 5 x 10-r* mzfsee. This value is up to about an order of magnitude higher than values calculated from the diffusion creep of alumina(B) or from initial sintering rates.(lO) The curvature of Fig. 7 is reduced if account is taken of closed porosity contributing to the overall pore fraction. If 4 per cent of the porosity is closed when the total porosity is 12 per cent then the plot in Fig. 7 becomes a straight line. It is important to consider only open interconnected porosity because closed pores will sinter at a different rate. If the closed pores contain gas the shrinkage rate may be very slow. CONCLUSEONS
The sinteing rate of equilibrium grain edge porosity may usefully be applied to the second stage sir&ring kinetics of powders. The model is valid between an upper and a lower threshold volume. Many powder compacts are expected to be below the upper threshold volume immediately after the formation of necks and the rounding of surfaces, Below the lower threshold volume the interconnected structure is unstable and porosity forms isolated pores. The sintering rate depends not only on the surface curvature but also on the grain boundary vacancy potential. At large pore volumes the boundary areas are relatively small and the vacancy oon~entration is greatly suppressed. As the pore volume decreases the boundary areas grow, the vacancy concentration increases and so the sintering force decreases, Simultaneously the pore surface curvature increases with decreasing pore volume. The increased curvature increases the local vacancy concentr&tion and at small pore volumes the sintering force increases. At intermediate pore volumes the sintering force passes through a minimum. During most of the second stage the grain boundary area is much smaller than the pore surface area. Co~equently the sintering rate is more sensitive to grain boundary area than pore area. The dihedral angle maintained between pore surface and grain boundary also affects sintering rate. If the angle is reduced to values observed in liquid precipitates the volume of precipitate grows rather than sinters.
BEERE:
The
quantitative
sintering
calculation
rates allows
the assessment data
SINTERING
a more
of
for straight
external
density.
atmosphere
Failure
When
coefficients relaxation
and
pore
to
interpreting
OF
where
POWDER
with
is relevant are
* = 0 dr
when
P(R) = Kay,
(15) R
27rRy, sin 8 = Equation
-
g
T/J(T) dr.
s0
(14) ensures that the vacancy
centre of the boundary
is zero.
the potential
on the pore surface.
may
rium concentration
many
ceramics
measurements
have
for
diffusion
creep
interfacial
kinetics may be rate controlling.
which
shown
that
is annihilated
equal to
If the boundary
maintains
of vacancies.
(15) fixes
an equilib-
When
a vacancy
the change in Gibbs free energy is zero.
Equation (16) meets this condition by balancing the change in chemical potential over the boundary
ACKNOWLEDGEMENT
This paper is published by permission Electricity Generating Board.
is a perfect sink the boundary
(16)
flux at the
Equation
It is also necessary for the grain boundaries to act as perfect sinks for vacancies. The last requirement for
of the central
with the work done on the external
restraint.
chemical potential
is
over the boundary
The
(17)
REFERENCES 1. J. WHITE, Sintering and Related Phenowaena, edited by KUCZYNSKI, p. 354. Gordon & Breach, New York (1967). 2. R. L. COBLE, J. uppl. Phys. 32, 787 (1961). 3. W. BEERE, J. Mat. Sci. 8, 1717 (1973). 4. W. BEERE, to be published. 5. M. 0. TUCKER and J. A. TURNBULL, Int. Conf. on Physical Met. of Reactor Fuel Ekments, Berkeley, England (1973). 6. C. S. SMITE, Met. Rev. 9, 1 (1964). W. OSTWALD, 2. Phys. Chena. 84, 495 (1900). :: R. L. COBLE.J. a~&. Phvs. 32. 793 (1961). 9. C. W. HEW&N c&d’ W. 0: K&&RY,‘ J. Ana. Ceram. Sot. 50, 218 (1967). 10. W. R. RAO and I. B. CUTLER, J. Am. Ceram. Sot. 55, 170 (1972). 11. B. BURTON and G. L. REYNOLDS, Acta Met. 21. 1073 (1973). APPENDIX
The sintering vacancy
rate is calculated
flux entering
by grain edge porosity.
where cc is a constant.
The flux at the pore surface
a grain boundary The boundary
deriving
the
and the total flux is J = j2rrRd. change of volume
Hence the rate of
(19) A tetrakaidecahedron by two grains. tion
porosity
has
14 faces
equation
is assumed
fraction
vacancy potential at a distance r from the boundary centre, then the potential satisfies the differential
each
A small change in the volume
AV/V
is equal
for a circle is equal to 2/R.
surrounded If p(r) is the
is
(18)
to
shared frac-
6(AV/ V) = 7 SU/
SXJ’% and the ratio of circumference by
to be circular, Fig. 1, and of radius R.
to area, L/A,
Incorporating
these into
(19) gives the rate of change of pore volume
dW/V -=-~(!%$j(~+K)(A+AA’)_ dt
equation
(20) The last term in brackets (13)
10
of
(14)
on the rim of the boundary
true
rate
r = 0
the potential
hold
to volumetric
surface
the
(13) is solved
changes.
not
is rapid compared
diffusion
that
on
Equation
subject to the conditions
increasing
when
such
depending
of vacancies.
it is
for this causes difficulty
sizes
145
COMPACTS
/l is a constant
annihilation
porosity. During not connected to
increases
to allow
in correct interpretation. The theory presented
stage
approach
line behaviour
important to consider only open sintering the fraction of porosity the
second
rigorous
of mechanisms.
and looking
KINETICS
(20) accounts
on the right of equation
for the reduction
the grains move together.
in pore volume
as