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Quasi-equilibrium sintering for coupled grain-boundary and surface diffusion
A c t a m e m l l , mater. Vol. 43, No. 2. pp. 499 506. 1995
~
Pergamon
0956-7151 (94)00249-5
Copyright ,!~ 1995 Elsevier Science Ltd Printed in Great Britain All rights reserved 0956-7151/95 $9.50 + 0.00
QUASI-EQUILIBRIUM SINTERING FOR COUPLED GRAIN-BOUNDARY AND SURFACE DIFFUSION J. SVOBODAt and H. RIEDEL Fraunhofer-lnstitut f/.ir Werkstoffmechanik, W6hlerstraBe l l, D-79108 Freiburg, Germany (Received 16 July 1993; in revised jorm 18 May 1994)
Abstract--Thesintering of a hexagonal array of wires is analysed taking into account grain-boundary
and surface diffusion. A numerical solution describes the whole process from initial neck formation to the stage in which pores have equilibrium, or near equilibrium shapes and finally disappear. The main purpose of the paper is to present an approximate analytical solution for the later stages of sintering. By taking into account the dissipation rate by surface diffusion the new solution extends the range of validity of the known equilibrium solution far into the range of low surface diffusivities and yet has a relatively simple form. To demonstrate the applicability to three-dimensional cases, an axisymmetric problem is also analysed.
1.
INTRODUCTION
longer duration of the first stage. After some time the pore shape globally resembles the equilibrium shape but still with large deviations near the edges of the grain boundaries, where the material is supplied to the pore surface. These deviations in surface curvature lead to a strong reduction of the sintering rate in the second stage. The coupled surface/grain boundary diffusion problem was analysed either numerically [3-10] or with approximate analytical estimates [11]. Both approaches are based on a detailed consideration of the surface curvature near the pore tips, which serves as a driving force for grain boundary diffusion. The idea proposed in the present paper is that in the second and third stages the coupled problem of surface and grain boundary diffusion can be accurately solved by a global approach in a much simpler way. We claim that the Gibbs free energy, G, and the dissipation rate, R, of the sintering system can be calculated by considering a sequence of equilibrium states, although the surface may be far out of equilibrium locally. The rate of the process is then given by the thermodynamic principle (~ + R = 0, where the superposed dot denotes the time derivative. We call this type of theory a quasi-equilibrium theory of stage 2 and stage 3 sintering. The resulting sintering rates arc compared with numerical calculations. For simplicity the idea is first demonstrated for the two-dimensional problem of a hexagonal array of sintering wires. Numerically, the whole process is simulated from the initial formation of a neck in stage 1 to full densification, similar to work reported in the literature, e.g. [3-10]. The aim is to show that alter a certain amount of shrinkage the pores assume quasi-equilibrium shapes (with possibly large deviations near the edge of the grain
The work described in this paper is part of a larger program which aims at the numerical simulation of powder-technological production steps. As far as sintering or hipping are concerned, one possible approach is to rely on purely phenomenological constitutive equations for the mechanical behavior [1,2]. The present authors believe that it is also worthwhile to exploit the information contained in existing sintering models for the formation of macroscopic constitutive equations. In some cases improvements of the models appear to be necessary and possible. It is useful to distinguish different stages of sintering. In our terminology, the first stage is characterized by isolated necks with the surface of the pore space being not yet in equilibrium, while in the second stage open porosity with (near-) equilibrium surfaces prevails, and the third stage is characterized by closed porosity, usually with equilibrium pore shapes. In the two-dimensional problem of sintering wires the distinction between the second and third stages is meaningless. The transport mechanism supporting densification is assumed to be grain boundary diffusion in the contact areas between particles. Surface diffusion serves to distribute the material across the surface of the pore space. If surface diffusion is fast, the pore surface assumes an equilibrium shape quickly, i.e. the first stage is short. Slow surface diffusion leads to a
?Permanent affiliation with Institute of Physics of Materials, Academy of Sciences of the Czech Republic, 61662 Brno, Zizkova 22, Czech Republic. 499
500
SVOBODA and RIEDEL:
SINTERING FOR GB AND SURFACE DIFFUSION
boundary), and that the quasi-equilibrium theory describes the numerical sintering rates accurately. To prove the applicability of the principle to three-dimensional configurations an axisymmetric diffusion problem is analysed, which is intended to simulate the growth of a circular neck and its interaction with its neighbors. Although the axisymmetric analysis involves an additional geometrical simplification in the quasi-equilibrium model, the agreement with the numerical model is good. A complete threedimensional model for surface diffusion on the doubly curved surfaces during the first and second stages of sintering is not yet available. Only equilibrium surfaces for the second stage have been constructed [12, 13]. 2. M O D E L
2.1. G o v e r n & g equations
The analysis is based on the phenomenological diffusion equations for the fluxes along the surface, j~, and along the grain boundary, Jb
Z-
27sf26Ds . -
vK
-
kT
(l)
~(~ D b
Jb=
kT
Va,.
(2)
Here 7~ and 7b are the surface and grain boundary energies per unit area, f~ is the atomic volume, 6 D S and ~D b are the transport coefficients for surface and grain boundary diffusion (in m3/s), k is the Boltzmann constant, T is the absolute temperature, V is the gradient operator, o., is the normal component of the stress acting on the grain boundary, and ~c is the mean curvature of the pore surface. For the twodimensional problems considered in the present paper ~c = l/(2p), where p is the radius of curvature. The requirement that matter is conserved gives the normal displacement rates of the pore surface, u~, and of the
a)
~m, £m
!
grains across the grain boundary due to the deposition of atoms, u, zi~-
27sf~6D~V2~c
(3)
kT
f l 6 D b V20"n
tin -
kT
(4)
"
The signs of the quantities are such that ~)s is positive when matter is removed from the surface, un is positive when the grains move apart, tensile stresses are positive, and the curvature is positive for a convex pore. 2.2. G e o m e t r y a n d b o u n d a r y conditions
Figure 1 shows the hexagonal array of cylindrical (two-dimensional) grains analysed in the present paper. Figure l(a) shows a possible starting configuration with circular cylinders; the shape of the pores between the grains is far from equilibrium. Figure l(b) shows the other extreme case, in which the pores have rounded, equilibrium shapes characterized by circular arcs connected by the dihedral angle 20, which is determined by cos 0 = 7h/(27~). An isostatic stress, o'm (equi-biaxial in two dimensions) is applied to the structure. Free sintering is characterized by a m= 0. The structure with hexagonal grains and equilibrium pore shapes was previously analysed by Hsueh and Evans [14] and by Riedel [15] assuming 6 Ds/c} Db --. o(3.
At the pore tip, equilibrium of the surface tension forces demands that the dihedral angle ~ is maintained. Continuity of the chemical potential requires that the stress on the grain boundary directly at the pore tip is related to the pore surface curvature at the tip by o.n,tip
27sgtip .
=
(5)
Conservation of matter demands that Jb,tip = 2L,,ip.
b)
(6)
Gm, ~m
!
11
C~m,~m
(Ym, EITI
t
!
Fig. 1. Hexagonal array of grains. (a) Initial configuration: circular cylinders. (b) Equilibrium pore shape.
SVOBODA and RIEDEL: SINTERING FOR GB AND SURFACE DIFFUSION 3. S O L U T I O N O F T H E G R A I N B O U N D A R Y DIFFUSION PROBLEM
For two-dimensional geometries with planar grain boundaries, the grain boundary diffusion equation (4) is readily solved, if the usual assumption of rigid grains is made. Then ti, does not vary along the grain boundary and one obtains
k TtJn
2
an = 2% X,ip+ ~ 6 ~ b (C -- X2)
(7)
where c is the half contact length (i.e. the half neck size) between adjacent grains. The associated flux is
(8)
Jb = - xC,~
where x is the coordinate along the grain boundary, with x = 0 in its center. To maintain global mechanical equilibrium the integral of a. over one grain boundary facet together with the surface tension forces at the pore tips must balance the applied stress
2k T 47~capc+3~bc3fi,+27~sinqJ
=o-md
(9)
where d is the distance between triple points [Fig. l(b)]. 4. N U M E R I C A L M E T H O D
Equation (4) was solved in closed form in the preceding section. Now the surface diffusion equation (3) is solved numerically using the finite difference method. The pore surface is subdivided by discrete points PI, P2 •. • P, as shown in Fig. 2 (only 1/6 of the pore surface needs to be considered because of the symmetry). Differentials at the knot points are approximated by parabolic interpolation, as described in [16].
(
(~m,Em
1 •
501
A possible starting confguration for the numerical analysis is similar to the configuration shown in Fig. 2. To start with even sharper pore tips causes numerical problems. The time integration is carried out by the explicit Euler method. If xi(t k) and y,(tk) are the coordinates of point P, on the pore surface at time tk, the coordinates at time tk +a= tk ÷ At are calculated as
xi(tk + 1) = X,(tk) -- Us(tk ))';(tk)At yi(tk+~)=yi(tk)+[fi~(tk)x;(tk)+i~n/2]At
(10) (11)
where the prime denotes the derivative with respect to the arc length and un is added to account for the rigid-body motion of the grains, when material is added to or removed from the horizontal grain boundary. Special attention is needed to satisfy the boundary conditions at the pore tip. At the beginning of each time step the curvature at the pore tip is calculated from the positions of the knot points P~ and P, and from the dihedral angle. Then the current t~. can be calculated from the equilibrium condition (9). Equation (8) gives Jb" On the other hand, .L is calculated numerically from equation (1) using the curvature at PI, P2 and P3. Generally the so calculated values of the fluxes do not satisfy the continuity equation (6). By adjusting the position of the pore tip (the point Pl) iteratively, however, one can achieve that equation (6) is fulfilled. 5. ANALYTICAL APPROXIMATION BY A QUASI-EQUILIBRIUM MODEL An approximate analytical solution is now developed not by explicit solution of the governing differential equation (1), but rather from the principle that the rate of change of the Gibbs free energy G must be equal to the negative dissipation rate R caused by the diffusive fluxes along the grain boundaries and along the pore surface [17] d + R = 0.
(12)
5.1. The two -dimensional model For the hexagonal array of wires the smallest possible unit cell comprises 1/12 of a grain and 1/6 of a pore. For this unit cell (~ is calculated from
Pn
(~ = _ a m/) +5~b " c" + "/~/' . ~
Om, Em C
.I\
Fig. 2. Nonequilibrium pore shape whcih can serve as the starting condition for the numerical calculations.
(13)
where V is the area of the unit cell (V = x/3d2/8), and L is 1/6 of the length of circumference of the pore. The factor 1/2 in front of the grain boundary energy term arises since the grain boundary is shared by two grains, so that only one half of it belongs to the unit cell. The dissipation rate is calculated from the fluxes by
kT f~ k T CL R - 2fft6Db j0 J ~dx + n6D~ J0 j ~ds
(14)
502
SVOBODA and RIEDEL: SINTERING FOR GB AND SURFACE DIFFUSION
where the factor 1/2 arises for the same reason as before, and s is the arc length along the pore surface measured from the pore tip. This principle is applied now to a pore with equilibrium shape. However, the analysis goes beyond the classical equilibrium theory [14, 15], since it takes into account not only the dissipation rate caused by grain boundary diffusion but also that caused by surface diffusion. This extends the validity of the result to very small ratios of 5D~/hDb, although the assumption of an equilibrium pore shape is generally thought to be linked to rapid surface diffusion, i.e. large 6D~/6D b ratios. For equilibrium pore shapes the structure is characterized uniquely by one state variable, which is chosen to be the relative density D. The geometrical quantities c, d, p (radius of curvature of the pore surface) and L are related to D by e~r
( ~ ')1/2[ 1 -- 24wf3(1 D sin 03(15) 6x/~D J L )/o 27
A =~-8
~Q-6
(16) (17)
Q = 30 - 2x//3 cos (k sin 0
(r~ = -7s + - ])b P x/3d"
(24)
The denominator 1 +ASDb/6D ~ in equation (23) describes the influence of a finite surface diffusion coefficient; A is given by
(25)
(20)
2D
~f3dgm.
(21)
L-s .L = -dG~ sin - P 2 ~pcos0sin
where
L -s] P
.
(26)
[15].)
b - - - -
,/3
with c from equation (15). Here ~r is the sintering stress of the equilibrium configuration
(19)
With these relationships, the rate of change of the Gibbs free energy and the dissipation rate can be expressed in terms of D a n d / ) . The only complicated step in the evaluation of equations (12)-(14) is the calculation of the surface flux, j~, which is associated with a densification rate/). The rigid-body motion of the grains, which is caused by the plating or removal of matter to or from the grain boundaries, would lead to nonequilibrium pore shapes, if it acted alone. To maintain the equilibrium shape, material must be deposited nonuniformly on the pore surface. The resulting flux is
(L-s)
O s
1 + A6Db/6D S
(18)
has the meaning of a dimensionless pore area (Qp2 is the pore area in physical dimensions). Further the isostatic strain rate and the normal displacement rate are related to the densification rate by
I
(7 m - -
(23)
2kTc 3
Equation (23) reproduces the result of the equilibrium model [14, 15] for 3Ds/3D b ~ oo, as it should. (There is a small difference in the sintering stress, compared to Ref. [15], since the contribution of the grain boundary energy was erroneously neglected in
where 0 = ~h - 72/6, and
--p
x/3fl6D~ -
2D
B = x/(1 - D )Q/x//3 •
~;2
L = pO
I~n =
19 -
[Q - 2x/3B sin 0 ]~
(rt(1-D))
~m =
~
0 3 + 4xf~(cos ~, + B)(0 cos 0 - sin 0 ) + (cos ~h + B)2(20 - sin 20 )
2~ x~1/2 d = r \ 3x/~ D j p =r
All geometrical quantities appearing in this equation were given in equations (16)-(18), and s is the arc length over which j~ is integrated in equation (14). The remaining steps to obtain the final result for the densification rate are: insert equations (8) and (12) into equation (14), carry out the integrals, replace V, c and L in equation (13) by D, and apply equation (12). By rearranging one arrives at the final result for the mean strain rate, or the densification rate
(22)
The correction term that accounts for the distribution of matter over the pore surface, ASDb/bDs, goes to zero as full density is approached, since B ---, 0 and A ---*0 for D ---, 1. For 0 = 60 ~ and D = 0.95 is A = 0.22, and the correction term may retard the densification substantially if (~Db/OD~ is large. Although one might first suspect that equation (23) is valid only as long as the correction term is small compared to 1, it turns out that it actually reproduces the numerical result over a wide range of conditions. 5.2. A geometrical simplification .for the two-dimensional model The detailed consideration of the geometrical conditions in the preceding section lead to a relatively complicated expression for A, which would become even more complicated in a three-dimensional case. Hence we introduce a geometrical simplification, verify its validity for the two-dimensional case and apply its analogue to a three-dimensional (axisymmetric) case in the next section. The arc length of the circular unit cell, before a neck develops, is ~zr/6. We assume that this total arc length remains constant during sintering and is shared by the neck, which occupies a length c, and the
SVOBODA and RIEDEL:
SINTERING FOR GB AND SURFACE DIFFUSION
dence on co is not strong, only this case is given here. The opposite case, co = 0 (i.e. a planar surface), was published previously [18]. Assuming a uniform plating of material on the surface, one can calculate the flux and the dissipation rate. The contribution by grain boundary diffusion on a circular neck was treated previously (e.g. [10, 13, 18]), and the result for the mean strain rate is found to be
r i i i
"L, EQUILIBRIUM~"
SURFACE
PARTICLE 1 ~1 I
/
//// __.
C
~
PARTICLE 2 Fig. 3. Axisymmetric model and simplified geometries (dashed lines). pore surface, which occupies the rest, r r r / 6 - c. The surface flux is assumed to decrease linearly from the pore tip to the end of the unit cell, i.e. material is deposited uniformly on the surface. With this flux the dissipation rate is evaluated easily. The result is the same as equation (23) but with A -
rc r
1
12c
2
gm -- ~rm -- ~ 3K
(27)
5.3. An axisymmetric model The eventual goal is to develop a model for threedimensional particle packings. As an approximation we consider an axisymmetric model with a circular neck (Fig. 3) between two initially spherical particles with radius r. Each particle has Z necks, so that the surface area on the sphere belonging to one neck is 47rr2/Z. This unit cell is shown in Fig. 3. We make the same geometrical assumption as in the preceding section, namely that this area is shared by the neck, ~c 2, and the free surface, 4~r2/Z - ;re 2. The shape of the surface is approximated by a conical surface (the dashed lines). The result becomes particularly simple if the cone angle is ~o = ~/2, i.e. when the surface is approximated by a circular cylinder. Since the depen-
(28)
with the bulk viscosity
D°'3Dl'3ZkTc4I K -
48f25Dbr
t~Db]
1+ A~
r2
Z
6c 2
24
(29)
(30)
The coordination number is typically Z = 8. The result is discussed in Section 6.5. 6. COMPARISON OF NUMERICAL AND ANALYTICAL RESULTS 6.1. Evolution of the pore shape Figure 4 shows three examples for the numerically calculated evolution of the pore shape. In Fig. 4(a) free sintering without a pressure is considered. Although the surface diffusion coefficient is assumed to be very small, the pore tends to assume an equilibrium shape quickly. This is understandable since material flows from the grain boundary onto the free surface through the pore tips. F r o m this (numerical) observation one would expect that the analytical approximation based on equilibrium pore shapes has a broad range of validity, a prediction which is substantiated later. Figure 4(b, c) show situations with an applied pressure. If the pressure is not too large, the pore shape is first distorted by the material that is forced into the pore at its tips, but later the
m
b)
j.
The first term arises from grain boundary diffusion, while the second accounts for surface diffusion. For the cylindrical surface model A -
This is very much simpler than equation (25). The two result agree numerically for a relative density D =0.91, while the approximate results is 12% smaller than the more accurate result for D = 0.98 (for tp = 60c).
a)
503
c)
Fig. 4. Evolution of pores for the combinations of parameters in Table 1 and for ~, = 60.
504
SVOBODA and RIEDEL: Table 1. Normalized pressure
Norm. time interval
qmr/7~
A / ~ O b3's/(kTr 4)
o~ 8
8.73 x l0 1.92 x 10 5 1.75 x 10 5
>.F-O0 O rr O 2
~D~/ODb
a b c
SINTERING FOR GB AND SURFACE DIFFUSION
0.001 0.1 0.1
10
0 26 69.3
pore assumes a near equilibrium shape, when the pore becomes small [Fig. 4(b)]. If the pressure is higher, the shape distortion is severe. The intrusions of material from the grain boundaries into the pore can even lead to a dissociation of the pore into three parts, as in Fig. 4(c).
i
I
i
I
~
I
~t = 60 ° ~Ds/6D b = 10 -3 e% - 0
numerical
46
equilibrium i
I
i
00
i
I
2.10 2
I
4.10 -2
NORMALIZED TIME
IX1
6.10 -2 t ~ 6 D b Ts k T r4
Fig. 7. Porosity, f = l - D , vs normalized time for 6D~/SD b = 0.001, °m = 0.
6.2. Dens(fication
Figures 5-8 show the evolution of the porosity, f = 1 - D , for different values of 5D~/6D b. Figures 5-7 describe pressureless sintering, while Fig. 8 shows a n example with an applied pressure. The numerical solutions start with n o n - e q u i l i b r i u m pore shapes. In all figures the full numerical solution is c o m p a r e d with the quasi-equilibrium estimate. The quasi-equilibrium solutions were o b t a i n e d by choosing the c o n s t a n t of integration such that the porosities agree with the numerically calculated ones at the longest times to which the numerical calculations were carried. It is a p p a r e n t that the quasi-equilibrium solution reproduces the numerical results nearly perfectly for longer times, even for very slow surface diffusion. To appreciate the influence of the correc10
~o >I-O
I
I
I
% numerical 8 ; ~.. ..... ~ "'"'..., 6 - I ........ 4 2
~/= 60 ° 8gs/~D b = l ~m=0
I~ ..,,,,, _........... equilibrium quasi, . ~ ......... (,SDs/6Db equmonum ~
0
%
,
.
I
I
I
2.10 -4
4.10 -4
6.10 .4
NORMALIZED
TIME
~"""-
8.10 -4
t D 8 D b 7s k T r4
Fig. 5. Porosity, f - I - - D , vs normalized time for 5D~/SDb = l and ~'~, a , , - 0. 10
~o >I-C~ O Q_
i
8 .
I
.
i
I
6 -
c% = 0
2
quasiequilibrium
0
~
J 4.10 -4
NORMALIZED
,
i
,
8-10 -4 TIME
The second sintering stage is reached when the contact size a n d the densification rate a p p r o a c h their equilibrium values. This occurs after a certain fraction t,/t~, of the time to sintering, or after a fraction (D, - D0)/(1 - Do) of the total a m o u n t of densification, where the subscript t indicates the transition. Table 2 shows the dependence of these fractions on 6D~/SD b for pure sintering. For slow surface diffusion, stage 1 contributes a b o u t 50% to the a m o u n t o f densification, which corresponds to a b o u t 20% of the (isothermal) sintering time. F o r 5 D s / a D b = 10, these fractions decrease to 5 a n d 1.5%, respectively.
Figure 9 shows the time to complete sintering, t~, starting from an initial density D 0 = 0 . 9 0 7 . The numerical calculations started either from the n o n e q u i l i b r i u m pore shape (dashed lines) or from the equilibrium pore shape (dotted line). The solid lines represent the sintering times calculated from the quasi-equilibrium model, e q u a t i o n (23), which neglects the enhanced shrinkage rates during initial
I
~ = 60 ° 8Ds/aD b = 0 . 1
,
6.3. Transition b e t w e e n stages 1 a n d 2
6.4. S i n t e r i n g times
~....~,
t3
tion term A a D b / O D ~ in e q u a t i o n (23) note the change of the time scale between Figs 6 and 7. The retardation of the densification rate by a b o u t two orders o f m a g n i t u d e is solely caused, a n d correctly described by the correction term.
[', 12.11 4
t ~ 6 D b Ys kT r4
Fig. 6. Porosity, f - 1 - D , vs normalized time for 6D~/FD b = 0.1, a,,, = 0.
10
so
8
>I--
6
©
2
I
I
I
~ = 60 ° 5Ds/SD b = 0.1 k
(~rn r/ys = -26 ""~
0
0
quasi-equilibrium
I
i
i
0.5"10 .4
1"10 4
1.5"10 4
NORMALIZED
Fig. 8. Porosity, f - 1 - D ,
TIME
"~.
2"10 .4
t ~ 6DbTs k T r4
vs normalized time for --267,/r.
,SD~/gJDb = O. 1, a,,1
- -
SVOBODA and RIEDEL: SINTERING FOR GB AND SURFACE DIFFUSION Table 2.
~D~/~D b (D,
Do)/(1
0.00l 0.5 0,2
Do)
t/t,
0.l 0.45 0.15
1 0.25 0.1
10 0.05 0.015
neck formation. Since this enhancement of the sintering rate is neglected, it is not surprising that the analytical results overestimate the sintering time. However, the analytical estimate agrees nearly perfectly with the numerical calculation that starts from an equilibrium pore shape. Only for the highest pressure is the approximation distinguishable from the numerical result (the dotted line). This good agreement with the numerical result shows that the estimate based on equilibrium pore shapes is accurate outside the initial transient even if 6Ds/6Db is as small as 0.001. In this range, the slope of the curve is - 1, i.e. the sintering time is t, cc 1/SD, as opposed to 1/6D b at 3Ds/SD b > 1. 6.5. Axisymmetric model
A numerical model for the axisymmetric case was worked out elsewhere [10]. The emphasis in that paper lies on neck growth in the first sintering stage characterized by nonequilibrium surfaces, but results can also be evaluated for equilibrium surfaces in the second stage. If this is done, one finds that equation (30) underestimates the numerical result by only 8% at D = 0.7 and by 20% at D = 0.9 (for t# = 60° and Z = 8). Since this accuracy suffices for all practical purposes, equation (30) appears to be a useful approximation for describing the effect of slow surface diffusivities on the sintering rate. 7. DISCUSSION The quasi-equilibrium model is a global approach to stage 2 sintering, when grain boundary and surface diffusion play a role, It is based on the Gibbs free energy and the dissipation rate, rather than on the consideration of local curvatures as in [3 11]. This leads to a very simple solution for the way in which grain boundary and surface diffusion cooperate in the second sintering stage [equations (23) and (29)]. I
I
10-1'
~
lo-2i
I
.~
I
I
equilibrium ==.'.==: numerical
I "',
c-, I.-
10 `3
If the temperature dependence of measured sintering rates in stage 2 is evaluated, one should keep in mind that both 5D~ and riDs can contribute to the rate and that either one of them, or a mixture of both can dominate as described by equation (23). Conversely if a sintering cycle is modeled, the temperature dependencies of 6D b and ~iD~ should be included. In the preceding section it was shown that the global approach (the quasi-equilibrium model) predicts sintering rates accurately compared to the numerical model even if 5D~/6D b is very small. This is true despite the fact that the local curvature near the pore tip may be far out of equilibrium; it approaches zero for small 6 D j 6 D u ratios and Or, = 0, and can even become negative for compressive stresses. Globally, however, the dissipation rate does not depend strongly on the local curvature, but rather depends on the overall diffusion length. Similarly the sintering stress, which is given by the change of the interfacial energies during densification, dG/dV, is determined by the global shape rather than by the local curvature. Limitations arise in the extreme case shown in Fig. 4(c), where the pore shape is driven so much out of equilibrium by the applied pressure that the approximation based on equilibrium shapes cannot be expected to provide reasonable estimates. Sintering is the inverse process to the growth of creep cavities under tensile stresses. At low stresses and high 6D~/SDb ratios cavities grow in an equilibrium mode, while in the opposite case they develop a crack-like shape [19, 20]. The intrusion of material into the pores under high compressive stresses [Fig. 4(c)] is the analogue to crack-like growth under high tensile stresses. If the intrusion problem were of practical interest, one could treat it mathematically quite in analogy to the crack-like growth problem [19, 20]; a preliminary analysis shows that wherever sin(t#/2) appears in equations (25)-(29) of [20], it is replaced by cos(t#/2). In agreement with our finding Chuang et al. [20] note that equilibrium growth predominates even at small 6Ds/6D~ ratios, if the stress is small. From their analysis one can infer that the stress range in which the quasi-equilibrium solution is valid increases in proportion to 6D~/SD b and is of the order of a few times the sintering stress if g)Ds/~D b = 1 [their equation (95)]. Chuang et al. [20] did not modify the equilibrium growth rate by the denominator 1 + A S D b / S D ~, as we do, but their similarity solutions contain the effect implicitly.
O'rnr/~s = 0
._~ 10. 3
1 0 .4
505
"'-,~"-"<------------~,~ " " ~ ; 10 -2
I 10 -1
I 1
crnr/% = -8.7 ~mr/% = -:26i 101
I 10 2
10 3
8Ds/SDb
Fig, 9. Normalized time to complete sintering v s &Ds/~Db calculated by integrating equation (23) (solid lines) and from the numerical solutions. The numerical solutions start from nonequilibrium pore shapes (dashed lines) or from equilibrium pores (dotted lines).
8. CONCLUSIONS Sintering of a hexagonal array of wires was modeled numerically and analytically taking into account surface and grain boundary diffusion. It is shown that after an initial period of non-equilibriumneck growth (stage 1), the global pore shape approaches an equilibrium state even for very small ratios of ~D~/~D b (stage 2). The sintering rate is retarded by slow surface diffusivities, possibly by orders of magnitude,
506
SVOBODA and RIEDEL:
SINTERING FOR GB AND SURFACE DIFFUSION
if 6 D s / 6 D b is small. The r e t a r d a t i o n is described accurately by a quasi-equilibrium model which includes the dissipation rate by surface diffusion and uses the sintering stress of a n equilibrium pore. The (inverse) densification rate in the second sintering stage has the simple additive form 1//5 zc 1/6D b + A/6D~ [equation (23)]. A n a n a l o g o u s quasi-equilibrium model for an axisymmetric neck approximately agrees with numerical results for the sintering rate. Hence it can be considered as a useful a p p r o x i m a t i o n for threedimensional configurations. Acknowledgement--The authors would like to thank the
Deutsche Forschungsgemeinschaft for financial support under contract Ri 329/18-1.
REFERENCES
l. M. Abouaf, J. L. Chenot, G. Raisson and B. Baudin, Int. J. num. ll,Iethods Engng 25, 191 (1988). 2. H. Riedel and D. Z. Sun, in Numerical Methods in Industrial Forming Processes, N U M I F O R M '92 (edited by J. L. Chenot, R. D. Wood and O. C. Zienkiewicz), p. 883. A. A. Balkema, Rotterdam (1992). 3. F. A. Nichols and W. W. Mullins, J. appl. Phys. 36, 1826 (1965). 4. F. A. Nichols, Aeta metall. 16, 103 (1968).
5. P. Bross and H. E. Exner, Acta metall. 27, 1013 (1979). 6. R. M. German and J. E. Lathrop, J. Mater. Sci. 13, 921 (1978). 7. H. E. Exner, Acta metall. 35, 587 (1987). 8. Y. Takahashi, F. Ueno and K. Nishiguchi, Acta metall. 36, 3007 (1988). 9. D. Bouvard and R. M. McMeeking, J. Am. Ceram. Soc. Submitted. 10. J. Svoboda and H. Riedel, Acta metall, mater. 42, 1 (1995). 11. F. B. Swinkels and M. F. Ashby, Acta metall. 29, 259 (1981). 12. J. Svoboda, H. Riedel and H. Zipse, Aeta metall, mater. 42, 435 (1994). 13. H. Riedel, H. Zipse and J. Svoboda, Aeta metall, mater. 42, 445 (1994). 14. C. H. Hsueh and A. G. Evans, Acta metall. 29, 1907 (1981). 15. H. Riedel, in Ceramic Transactions, Vol. 12: Ceramic Powder Science l l l (edited by G. L. Messing, S.-I. Hirano and H. Hausner), pp. 619~30. Am. Ceram. Soc., Westerville, Ohio (1990). 16. J. Svoboda and H. Riedel, Acta metall, mater. 40, 2829 (1992). 17. J. Svoboda and I. Turek, Phil. Mag. B 64, 749 (1991). 18. H. Riedel, J. Svoboda and H. Zipse, in P M '94 (edited by D. Francois), Vol. 1, p. 663. Les Editions de Physique, Les Ulis, France (1994). 19. T.-J. Chuang and J. R. Rice, Acta metall. 21, 1625 (1973). 20. T.-J. Chuang, K. I. Kagawa, J. R. Rice and L. B. Sills, Acta metall. 27, 265 (1979).
~
Pergamon
0956-7151 (94)00249-5
Copyright ,!~ 1995 Elsevier Science Ltd Printed in Great Britain All rights reserved 0956-7151/95 $9.50 + 0.00
QUASI-EQUILIBRIUM SINTERING FOR COUPLED GRAIN-BOUNDARY AND SURFACE DIFFUSION J. SVOBODAt and H. RIEDEL Fraunhofer-lnstitut f/.ir Werkstoffmechanik, W6hlerstraBe l l, D-79108 Freiburg, Germany (Received 16 July 1993; in revised jorm 18 May 1994)
Abstract--Thesintering of a hexagonal array of wires is analysed taking into account grain-boundary
and surface diffusion. A numerical solution describes the whole process from initial neck formation to the stage in which pores have equilibrium, or near equilibrium shapes and finally disappear. The main purpose of the paper is to present an approximate analytical solution for the later stages of sintering. By taking into account the dissipation rate by surface diffusion the new solution extends the range of validity of the known equilibrium solution far into the range of low surface diffusivities and yet has a relatively simple form. To demonstrate the applicability to three-dimensional cases, an axisymmetric problem is also analysed.
1.
INTRODUCTION
longer duration of the first stage. After some time the pore shape globally resembles the equilibrium shape but still with large deviations near the edges of the grain boundaries, where the material is supplied to the pore surface. These deviations in surface curvature lead to a strong reduction of the sintering rate in the second stage. The coupled surface/grain boundary diffusion problem was analysed either numerically [3-10] or with approximate analytical estimates [11]. Both approaches are based on a detailed consideration of the surface curvature near the pore tips, which serves as a driving force for grain boundary diffusion. The idea proposed in the present paper is that in the second and third stages the coupled problem of surface and grain boundary diffusion can be accurately solved by a global approach in a much simpler way. We claim that the Gibbs free energy, G, and the dissipation rate, R, of the sintering system can be calculated by considering a sequence of equilibrium states, although the surface may be far out of equilibrium locally. The rate of the process is then given by the thermodynamic principle (~ + R = 0, where the superposed dot denotes the time derivative. We call this type of theory a quasi-equilibrium theory of stage 2 and stage 3 sintering. The resulting sintering rates arc compared with numerical calculations. For simplicity the idea is first demonstrated for the two-dimensional problem of a hexagonal array of sintering wires. Numerically, the whole process is simulated from the initial formation of a neck in stage 1 to full densification, similar to work reported in the literature, e.g. [3-10]. The aim is to show that alter a certain amount of shrinkage the pores assume quasi-equilibrium shapes (with possibly large deviations near the edge of the grain
The work described in this paper is part of a larger program which aims at the numerical simulation of powder-technological production steps. As far as sintering or hipping are concerned, one possible approach is to rely on purely phenomenological constitutive equations for the mechanical behavior [1,2]. The present authors believe that it is also worthwhile to exploit the information contained in existing sintering models for the formation of macroscopic constitutive equations. In some cases improvements of the models appear to be necessary and possible. It is useful to distinguish different stages of sintering. In our terminology, the first stage is characterized by isolated necks with the surface of the pore space being not yet in equilibrium, while in the second stage open porosity with (near-) equilibrium surfaces prevails, and the third stage is characterized by closed porosity, usually with equilibrium pore shapes. In the two-dimensional problem of sintering wires the distinction between the second and third stages is meaningless. The transport mechanism supporting densification is assumed to be grain boundary diffusion in the contact areas between particles. Surface diffusion serves to distribute the material across the surface of the pore space. If surface diffusion is fast, the pore surface assumes an equilibrium shape quickly, i.e. the first stage is short. Slow surface diffusion leads to a
?Permanent affiliation with Institute of Physics of Materials, Academy of Sciences of the Czech Republic, 61662 Brno, Zizkova 22, Czech Republic. 499
500
SVOBODA and RIEDEL:
SINTERING FOR GB AND SURFACE DIFFUSION
boundary), and that the quasi-equilibrium theory describes the numerical sintering rates accurately. To prove the applicability of the principle to three-dimensional configurations an axisymmetric diffusion problem is analysed, which is intended to simulate the growth of a circular neck and its interaction with its neighbors. Although the axisymmetric analysis involves an additional geometrical simplification in the quasi-equilibrium model, the agreement with the numerical model is good. A complete threedimensional model for surface diffusion on the doubly curved surfaces during the first and second stages of sintering is not yet available. Only equilibrium surfaces for the second stage have been constructed [12, 13]. 2. M O D E L
2.1. G o v e r n & g equations
The analysis is based on the phenomenological diffusion equations for the fluxes along the surface, j~, and along the grain boundary, Jb
Z-
27sf26Ds . -
vK
-
kT
(l)
~(~ D b
Jb=
kT
Va,.
(2)
Here 7~ and 7b are the surface and grain boundary energies per unit area, f~ is the atomic volume, 6 D S and ~D b are the transport coefficients for surface and grain boundary diffusion (in m3/s), k is the Boltzmann constant, T is the absolute temperature, V is the gradient operator, o., is the normal component of the stress acting on the grain boundary, and ~c is the mean curvature of the pore surface. For the twodimensional problems considered in the present paper ~c = l/(2p), where p is the radius of curvature. The requirement that matter is conserved gives the normal displacement rates of the pore surface, u~, and of the
a)
~m, £m
!
grains across the grain boundary due to the deposition of atoms, u, zi~-
27sf~6D~V2~c
(3)
kT
f l 6 D b V20"n
tin -
kT
(4)
"
The signs of the quantities are such that ~)s is positive when matter is removed from the surface, un is positive when the grains move apart, tensile stresses are positive, and the curvature is positive for a convex pore. 2.2. G e o m e t r y a n d b o u n d a r y conditions
Figure 1 shows the hexagonal array of cylindrical (two-dimensional) grains analysed in the present paper. Figure l(a) shows a possible starting configuration with circular cylinders; the shape of the pores between the grains is far from equilibrium. Figure l(b) shows the other extreme case, in which the pores have rounded, equilibrium shapes characterized by circular arcs connected by the dihedral angle 20, which is determined by cos 0 = 7h/(27~). An isostatic stress, o'm (equi-biaxial in two dimensions) is applied to the structure. Free sintering is characterized by a m= 0. The structure with hexagonal grains and equilibrium pore shapes was previously analysed by Hsueh and Evans [14] and by Riedel [15] assuming 6 Ds/c} Db --. o(3.
At the pore tip, equilibrium of the surface tension forces demands that the dihedral angle ~ is maintained. Continuity of the chemical potential requires that the stress on the grain boundary directly at the pore tip is related to the pore surface curvature at the tip by o.n,tip
27sgtip .
=
(5)
Conservation of matter demands that Jb,tip = 2L,,ip.
b)
(6)
Gm, ~m
!
11
C~m,~m
(Ym, EITI
t
!
Fig. 1. Hexagonal array of grains. (a) Initial configuration: circular cylinders. (b) Equilibrium pore shape.
SVOBODA and RIEDEL: SINTERING FOR GB AND SURFACE DIFFUSION 3. S O L U T I O N O F T H E G R A I N B O U N D A R Y DIFFUSION PROBLEM
For two-dimensional geometries with planar grain boundaries, the grain boundary diffusion equation (4) is readily solved, if the usual assumption of rigid grains is made. Then ti, does not vary along the grain boundary and one obtains
k TtJn
2
an = 2% X,ip+ ~ 6 ~ b (C -- X2)
(7)
where c is the half contact length (i.e. the half neck size) between adjacent grains. The associated flux is
(8)
Jb = - xC,~
where x is the coordinate along the grain boundary, with x = 0 in its center. To maintain global mechanical equilibrium the integral of a. over one grain boundary facet together with the surface tension forces at the pore tips must balance the applied stress
2k T 47~capc+3~bc3fi,+27~sinqJ
=o-md
(9)
where d is the distance between triple points [Fig. l(b)]. 4. N U M E R I C A L M E T H O D
Equation (4) was solved in closed form in the preceding section. Now the surface diffusion equation (3) is solved numerically using the finite difference method. The pore surface is subdivided by discrete points PI, P2 •. • P, as shown in Fig. 2 (only 1/6 of the pore surface needs to be considered because of the symmetry). Differentials at the knot points are approximated by parabolic interpolation, as described in [16].
(
(~m,Em
1 •
501
A possible starting confguration for the numerical analysis is similar to the configuration shown in Fig. 2. To start with even sharper pore tips causes numerical problems. The time integration is carried out by the explicit Euler method. If xi(t k) and y,(tk) are the coordinates of point P, on the pore surface at time tk, the coordinates at time tk +a= tk ÷ At are calculated as
xi(tk + 1) = X,(tk) -- Us(tk ))';(tk)At yi(tk+~)=yi(tk)+[fi~(tk)x;(tk)+i~n/2]At
(10) (11)
where the prime denotes the derivative with respect to the arc length and un is added to account for the rigid-body motion of the grains, when material is added to or removed from the horizontal grain boundary. Special attention is needed to satisfy the boundary conditions at the pore tip. At the beginning of each time step the curvature at the pore tip is calculated from the positions of the knot points P~ and P, and from the dihedral angle. Then the current t~. can be calculated from the equilibrium condition (9). Equation (8) gives Jb" On the other hand, .L is calculated numerically from equation (1) using the curvature at PI, P2 and P3. Generally the so calculated values of the fluxes do not satisfy the continuity equation (6). By adjusting the position of the pore tip (the point Pl) iteratively, however, one can achieve that equation (6) is fulfilled. 5. ANALYTICAL APPROXIMATION BY A QUASI-EQUILIBRIUM MODEL An approximate analytical solution is now developed not by explicit solution of the governing differential equation (1), but rather from the principle that the rate of change of the Gibbs free energy G must be equal to the negative dissipation rate R caused by the diffusive fluxes along the grain boundaries and along the pore surface [17] d + R = 0.
(12)
5.1. The two -dimensional model For the hexagonal array of wires the smallest possible unit cell comprises 1/12 of a grain and 1/6 of a pore. For this unit cell (~ is calculated from
Pn
(~ = _ a m/) +5~b " c" + "/~/' . ~
Om, Em C
.I\
Fig. 2. Nonequilibrium pore shape whcih can serve as the starting condition for the numerical calculations.
(13)
where V is the area of the unit cell (V = x/3d2/8), and L is 1/6 of the length of circumference of the pore. The factor 1/2 in front of the grain boundary energy term arises since the grain boundary is shared by two grains, so that only one half of it belongs to the unit cell. The dissipation rate is calculated from the fluxes by
kT f~ k T CL R - 2fft6Db j0 J ~dx + n6D~ J0 j ~ds
(14)
502
SVOBODA and RIEDEL: SINTERING FOR GB AND SURFACE DIFFUSION
where the factor 1/2 arises for the same reason as before, and s is the arc length along the pore surface measured from the pore tip. This principle is applied now to a pore with equilibrium shape. However, the analysis goes beyond the classical equilibrium theory [14, 15], since it takes into account not only the dissipation rate caused by grain boundary diffusion but also that caused by surface diffusion. This extends the validity of the result to very small ratios of 5D~/hDb, although the assumption of an equilibrium pore shape is generally thought to be linked to rapid surface diffusion, i.e. large 6D~/6D b ratios. For equilibrium pore shapes the structure is characterized uniquely by one state variable, which is chosen to be the relative density D. The geometrical quantities c, d, p (radius of curvature of the pore surface) and L are related to D by e~r
( ~ ')1/2[ 1 -- 24wf3(1 D sin 03(15) 6x/~D J L )/o 27
A =~-8
~Q-6
(16) (17)
Q = 30 - 2x//3 cos (k sin 0
(r~ = -7s + - ])b P x/3d"
(24)
The denominator 1 +ASDb/6D ~ in equation (23) describes the influence of a finite surface diffusion coefficient; A is given by
(25)
(20)
2D
~f3dgm.
(21)
L-s .L = -dG~ sin - P 2 ~pcos0sin
where
L -s] P
.
(26)
[15].)
b - - - -
,/3
with c from equation (15). Here ~r is the sintering stress of the equilibrium configuration
(19)
With these relationships, the rate of change of the Gibbs free energy and the dissipation rate can be expressed in terms of D a n d / ) . The only complicated step in the evaluation of equations (12)-(14) is the calculation of the surface flux, j~, which is associated with a densification rate/). The rigid-body motion of the grains, which is caused by the plating or removal of matter to or from the grain boundaries, would lead to nonequilibrium pore shapes, if it acted alone. To maintain the equilibrium shape, material must be deposited nonuniformly on the pore surface. The resulting flux is
(L-s)
O s
1 + A6Db/6D S
(18)
has the meaning of a dimensionless pore area (Qp2 is the pore area in physical dimensions). Further the isostatic strain rate and the normal displacement rate are related to the densification rate by
I
(7 m - -
(23)
2kTc 3
Equation (23) reproduces the result of the equilibrium model [14, 15] for 3Ds/3D b ~ oo, as it should. (There is a small difference in the sintering stress, compared to Ref. [15], since the contribution of the grain boundary energy was erroneously neglected in
where 0 = ~h - 72/6, and
--p
x/3fl6D~ -
2D
B = x/(1 - D )Q/x//3 •
~;2
L = pO
I~n =
19 -
[Q - 2x/3B sin 0 ]~
(rt(1-D))
~m =
~
0 3 + 4xf~(cos ~, + B)(0 cos 0 - sin 0 ) + (cos ~h + B)2(20 - sin 20 )
2~ x~1/2 d = r \ 3x/~ D j p =r
All geometrical quantities appearing in this equation were given in equations (16)-(18), and s is the arc length over which j~ is integrated in equation (14). The remaining steps to obtain the final result for the densification rate are: insert equations (8) and (12) into equation (14), carry out the integrals, replace V, c and L in equation (13) by D, and apply equation (12). By rearranging one arrives at the final result for the mean strain rate, or the densification rate
(22)
The correction term that accounts for the distribution of matter over the pore surface, ASDb/bDs, goes to zero as full density is approached, since B ---, 0 and A ---*0 for D ---, 1. For 0 = 60 ~ and D = 0.95 is A = 0.22, and the correction term may retard the densification substantially if (~Db/OD~ is large. Although one might first suspect that equation (23) is valid only as long as the correction term is small compared to 1, it turns out that it actually reproduces the numerical result over a wide range of conditions. 5.2. A geometrical simplification .for the two-dimensional model The detailed consideration of the geometrical conditions in the preceding section lead to a relatively complicated expression for A, which would become even more complicated in a three-dimensional case. Hence we introduce a geometrical simplification, verify its validity for the two-dimensional case and apply its analogue to a three-dimensional (axisymmetric) case in the next section. The arc length of the circular unit cell, before a neck develops, is ~zr/6. We assume that this total arc length remains constant during sintering and is shared by the neck, which occupies a length c, and the
SVOBODA and RIEDEL:
SINTERING FOR GB AND SURFACE DIFFUSION
dence on co is not strong, only this case is given here. The opposite case, co = 0 (i.e. a planar surface), was published previously [18]. Assuming a uniform plating of material on the surface, one can calculate the flux and the dissipation rate. The contribution by grain boundary diffusion on a circular neck was treated previously (e.g. [10, 13, 18]), and the result for the mean strain rate is found to be
r i i i
"L, EQUILIBRIUM~"
SURFACE
PARTICLE 1 ~1 I
/
//// __.
C
~
PARTICLE 2 Fig. 3. Axisymmetric model and simplified geometries (dashed lines). pore surface, which occupies the rest, r r r / 6 - c. The surface flux is assumed to decrease linearly from the pore tip to the end of the unit cell, i.e. material is deposited uniformly on the surface. With this flux the dissipation rate is evaluated easily. The result is the same as equation (23) but with A -
rc r
1
12c
2
gm -- ~rm -- ~ 3K
(27)
5.3. An axisymmetric model The eventual goal is to develop a model for threedimensional particle packings. As an approximation we consider an axisymmetric model with a circular neck (Fig. 3) between two initially spherical particles with radius r. Each particle has Z necks, so that the surface area on the sphere belonging to one neck is 47rr2/Z. This unit cell is shown in Fig. 3. We make the same geometrical assumption as in the preceding section, namely that this area is shared by the neck, ~c 2, and the free surface, 4~r2/Z - ;re 2. The shape of the surface is approximated by a conical surface (the dashed lines). The result becomes particularly simple if the cone angle is ~o = ~/2, i.e. when the surface is approximated by a circular cylinder. Since the depen-
(28)
with the bulk viscosity
D°'3Dl'3ZkTc4I K -
48f25Dbr
t~Db]
1+ A~
r2
Z
6c 2
24
(29)
(30)
The coordination number is typically Z = 8. The result is discussed in Section 6.5. 6. COMPARISON OF NUMERICAL AND ANALYTICAL RESULTS 6.1. Evolution of the pore shape Figure 4 shows three examples for the numerically calculated evolution of the pore shape. In Fig. 4(a) free sintering without a pressure is considered. Although the surface diffusion coefficient is assumed to be very small, the pore tends to assume an equilibrium shape quickly. This is understandable since material flows from the grain boundary onto the free surface through the pore tips. F r o m this (numerical) observation one would expect that the analytical approximation based on equilibrium pore shapes has a broad range of validity, a prediction which is substantiated later. Figure 4(b, c) show situations with an applied pressure. If the pressure is not too large, the pore shape is first distorted by the material that is forced into the pore at its tips, but later the
m
b)
j.
The first term arises from grain boundary diffusion, while the second accounts for surface diffusion. For the cylindrical surface model A -
This is very much simpler than equation (25). The two result agree numerically for a relative density D =0.91, while the approximate results is 12% smaller than the more accurate result for D = 0.98 (for tp = 60c).
a)
503
c)
Fig. 4. Evolution of pores for the combinations of parameters in Table 1 and for ~, = 60.
504
SVOBODA and RIEDEL: Table 1. Normalized pressure
Norm. time interval
qmr/7~
A / ~ O b3's/(kTr 4)
o~ 8
8.73 x l0 1.92 x 10 5 1.75 x 10 5
>.F-O0 O rr O 2
~D~/ODb
a b c
SINTERING FOR GB AND SURFACE DIFFUSION
0.001 0.1 0.1
10
0 26 69.3
pore assumes a near equilibrium shape, when the pore becomes small [Fig. 4(b)]. If the pressure is higher, the shape distortion is severe. The intrusions of material from the grain boundaries into the pore can even lead to a dissociation of the pore into three parts, as in Fig. 4(c).
i
I
i
I
~
I
~t = 60 ° ~Ds/6D b = 10 -3 e% - 0
numerical
46
equilibrium i
I
i
00
i
I
2.10 2
I
4.10 -2
NORMALIZED TIME
IX1
6.10 -2 t ~ 6 D b Ts k T r4
Fig. 7. Porosity, f = l - D , vs normalized time for 6D~/SD b = 0.001, °m = 0.
6.2. Dens(fication
Figures 5-8 show the evolution of the porosity, f = 1 - D , for different values of 5D~/6D b. Figures 5-7 describe pressureless sintering, while Fig. 8 shows a n example with an applied pressure. The numerical solutions start with n o n - e q u i l i b r i u m pore shapes. In all figures the full numerical solution is c o m p a r e d with the quasi-equilibrium estimate. The quasi-equilibrium solutions were o b t a i n e d by choosing the c o n s t a n t of integration such that the porosities agree with the numerically calculated ones at the longest times to which the numerical calculations were carried. It is a p p a r e n t that the quasi-equilibrium solution reproduces the numerical results nearly perfectly for longer times, even for very slow surface diffusion. To appreciate the influence of the correc10
~o >I-O
I
I
I
% numerical 8 ; ~.. ..... ~ "'"'..., 6 - I ........ 4 2
~/= 60 ° 8gs/~D b = l ~m=0
I~ ..,,,,, _........... equilibrium quasi, . ~ ......... (,SDs/6Db equmonum ~
0
%
,
.
I
I
I
2.10 -4
4.10 -4
6.10 .4
NORMALIZED
TIME
~"""-
8.10 -4
t D 8 D b 7s k T r4
Fig. 5. Porosity, f - I - - D , vs normalized time for 5D~/SDb = l and ~'~, a , , - 0. 10
~o >I-C~ O Q_
i
8 .
I
.
i
I
6 -
c% = 0
2
quasiequilibrium
0
~
J 4.10 -4
NORMALIZED
,
i
,
8-10 -4 TIME
The second sintering stage is reached when the contact size a n d the densification rate a p p r o a c h their equilibrium values. This occurs after a certain fraction t,/t~, of the time to sintering, or after a fraction (D, - D0)/(1 - Do) of the total a m o u n t of densification, where the subscript t indicates the transition. Table 2 shows the dependence of these fractions on 6D~/SD b for pure sintering. For slow surface diffusion, stage 1 contributes a b o u t 50% to the a m o u n t o f densification, which corresponds to a b o u t 20% of the (isothermal) sintering time. F o r 5 D s / a D b = 10, these fractions decrease to 5 a n d 1.5%, respectively.
Figure 9 shows the time to complete sintering, t~, starting from an initial density D 0 = 0 . 9 0 7 . The numerical calculations started either from the n o n e q u i l i b r i u m pore shape (dashed lines) or from the equilibrium pore shape (dotted line). The solid lines represent the sintering times calculated from the quasi-equilibrium model, e q u a t i o n (23), which neglects the enhanced shrinkage rates during initial
I
~ = 60 ° 8Ds/aD b = 0 . 1
,
6.3. Transition b e t w e e n stages 1 a n d 2
6.4. S i n t e r i n g times
~....~,
t3
tion term A a D b / O D ~ in e q u a t i o n (23) note the change of the time scale between Figs 6 and 7. The retardation of the densification rate by a b o u t two orders o f m a g n i t u d e is solely caused, a n d correctly described by the correction term.
[', 12.11 4
t ~ 6 D b Ys kT r4
Fig. 6. Porosity, f - 1 - D , vs normalized time for 6D~/FD b = 0.1, a,,, = 0.
10
so
8
>I--
6
©
2
I
I
I
~ = 60 ° 5Ds/SD b = 0.1 k
(~rn r/ys = -26 ""~
0
0
quasi-equilibrium
I
i
i
0.5"10 .4
1"10 4
1.5"10 4
NORMALIZED
Fig. 8. Porosity, f - 1 - D ,
TIME
"~.
2"10 .4
t ~ 6DbTs k T r4
vs normalized time for --267,/r.
,SD~/gJDb = O. 1, a,,1
- -
SVOBODA and RIEDEL: SINTERING FOR GB AND SURFACE DIFFUSION Table 2.
~D~/~D b (D,
Do)/(1
0.00l 0.5 0,2
Do)
t/t,
0.l 0.45 0.15
1 0.25 0.1
10 0.05 0.015
neck formation. Since this enhancement of the sintering rate is neglected, it is not surprising that the analytical results overestimate the sintering time. However, the analytical estimate agrees nearly perfectly with the numerical calculation that starts from an equilibrium pore shape. Only for the highest pressure is the approximation distinguishable from the numerical result (the dotted line). This good agreement with the numerical result shows that the estimate based on equilibrium pore shapes is accurate outside the initial transient even if 6Ds/6Db is as small as 0.001. In this range, the slope of the curve is - 1, i.e. the sintering time is t, cc 1/SD, as opposed to 1/6D b at 3Ds/SD b > 1. 6.5. Axisymmetric model
A numerical model for the axisymmetric case was worked out elsewhere [10]. The emphasis in that paper lies on neck growth in the first sintering stage characterized by nonequilibrium surfaces, but results can also be evaluated for equilibrium surfaces in the second stage. If this is done, one finds that equation (30) underestimates the numerical result by only 8% at D = 0.7 and by 20% at D = 0.9 (for t# = 60° and Z = 8). Since this accuracy suffices for all practical purposes, equation (30) appears to be a useful approximation for describing the effect of slow surface diffusivities on the sintering rate. 7. DISCUSSION The quasi-equilibrium model is a global approach to stage 2 sintering, when grain boundary and surface diffusion play a role, It is based on the Gibbs free energy and the dissipation rate, rather than on the consideration of local curvatures as in [3 11]. This leads to a very simple solution for the way in which grain boundary and surface diffusion cooperate in the second sintering stage [equations (23) and (29)]. I
I
10-1'
~
lo-2i
I
.~
I
I
equilibrium ==.'.==: numerical
I "',
c-, I.-
10 `3
If the temperature dependence of measured sintering rates in stage 2 is evaluated, one should keep in mind that both 5D~ and riDs can contribute to the rate and that either one of them, or a mixture of both can dominate as described by equation (23). Conversely if a sintering cycle is modeled, the temperature dependencies of 6D b and ~iD~ should be included. In the preceding section it was shown that the global approach (the quasi-equilibrium model) predicts sintering rates accurately compared to the numerical model even if 5D~/6D b is very small. This is true despite the fact that the local curvature near the pore tip may be far out of equilibrium; it approaches zero for small 6 D j 6 D u ratios and Or, = 0, and can even become negative for compressive stresses. Globally, however, the dissipation rate does not depend strongly on the local curvature, but rather depends on the overall diffusion length. Similarly the sintering stress, which is given by the change of the interfacial energies during densification, dG/dV, is determined by the global shape rather than by the local curvature. Limitations arise in the extreme case shown in Fig. 4(c), where the pore shape is driven so much out of equilibrium by the applied pressure that the approximation based on equilibrium shapes cannot be expected to provide reasonable estimates. Sintering is the inverse process to the growth of creep cavities under tensile stresses. At low stresses and high 6D~/SDb ratios cavities grow in an equilibrium mode, while in the opposite case they develop a crack-like shape [19, 20]. The intrusion of material into the pores under high compressive stresses [Fig. 4(c)] is the analogue to crack-like growth under high tensile stresses. If the intrusion problem were of practical interest, one could treat it mathematically quite in analogy to the crack-like growth problem [19, 20]; a preliminary analysis shows that wherever sin(t#/2) appears in equations (25)-(29) of [20], it is replaced by cos(t#/2). In agreement with our finding Chuang et al. [20] note that equilibrium growth predominates even at small 6Ds/6D~ ratios, if the stress is small. From their analysis one can infer that the stress range in which the quasi-equilibrium solution is valid increases in proportion to 6D~/SD b and is of the order of a few times the sintering stress if g)Ds/~D b = 1 [their equation (95)]. Chuang et al. [20] did not modify the equilibrium growth rate by the denominator 1 + A S D b / S D ~, as we do, but their similarity solutions contain the effect implicitly.
O'rnr/~s = 0
._~ 10. 3
1 0 .4
505
"'-,~"-"<------------~,~ " " ~ ; 10 -2
I 10 -1
I 1
crnr/% = -8.7 ~mr/% = -:26i 101
I 10 2
10 3
8Ds/SDb
Fig, 9. Normalized time to complete sintering v s &Ds/~Db calculated by integrating equation (23) (solid lines) and from the numerical solutions. The numerical solutions start from nonequilibrium pore shapes (dashed lines) or from equilibrium pores (dotted lines).
8. CONCLUSIONS Sintering of a hexagonal array of wires was modeled numerically and analytically taking into account surface and grain boundary diffusion. It is shown that after an initial period of non-equilibriumneck growth (stage 1), the global pore shape approaches an equilibrium state even for very small ratios of ~D~/~D b (stage 2). The sintering rate is retarded by slow surface diffusivities, possibly by orders of magnitude,
506
SVOBODA and RIEDEL:
SINTERING FOR GB AND SURFACE DIFFUSION
if 6 D s / 6 D b is small. The r e t a r d a t i o n is described accurately by a quasi-equilibrium model which includes the dissipation rate by surface diffusion and uses the sintering stress of a n equilibrium pore. The (inverse) densification rate in the second sintering stage has the simple additive form 1//5 zc 1/6D b + A/6D~ [equation (23)]. A n a n a l o g o u s quasi-equilibrium model for an axisymmetric neck approximately agrees with numerical results for the sintering rate. Hence it can be considered as a useful a p p r o x i m a t i o n for threedimensional configurations. Acknowledgement--The authors would like to thank the
Deutsche Forschungsgemeinschaft for financial support under contract Ri 329/18-1.
REFERENCES
l. M. Abouaf, J. L. Chenot, G. Raisson and B. Baudin, Int. J. num. ll,Iethods Engng 25, 191 (1988). 2. H. Riedel and D. Z. Sun, in Numerical Methods in Industrial Forming Processes, N U M I F O R M '92 (edited by J. L. Chenot, R. D. Wood and O. C. Zienkiewicz), p. 883. A. A. Balkema, Rotterdam (1992). 3. F. A. Nichols and W. W. Mullins, J. appl. Phys. 36, 1826 (1965). 4. F. A. Nichols, Aeta metall. 16, 103 (1968).
5. P. Bross and H. E. Exner, Acta metall. 27, 1013 (1979). 6. R. M. German and J. E. Lathrop, J. Mater. Sci. 13, 921 (1978). 7. H. E. Exner, Acta metall. 35, 587 (1987). 8. Y. Takahashi, F. Ueno and K. Nishiguchi, Acta metall. 36, 3007 (1988). 9. D. Bouvard and R. M. McMeeking, J. Am. Ceram. Soc. Submitted. 10. J. Svoboda and H. Riedel, Acta metall, mater. 42, 1 (1995). 11. F. B. Swinkels and M. F. Ashby, Acta metall. 29, 259 (1981). 12. J. Svoboda, H. Riedel and H. Zipse, Aeta metall, mater. 42, 435 (1994). 13. H. Riedel, H. Zipse and J. Svoboda, Aeta metall, mater. 42, 445 (1994). 14. C. H. Hsueh and A. G. Evans, Acta metall. 29, 1907 (1981). 15. H. Riedel, in Ceramic Transactions, Vol. 12: Ceramic Powder Science l l l (edited by G. L. Messing, S.-I. Hirano and H. Hausner), pp. 619~30. Am. Ceram. Soc., Westerville, Ohio (1990). 16. J. Svoboda and H. Riedel, Acta metall, mater. 40, 2829 (1992). 17. J. Svoboda and I. Turek, Phil. Mag. B 64, 749 (1991). 18. H. Riedel, J. Svoboda and H. Zipse, in P M '94 (edited by D. Francois), Vol. 1, p. 663. Les Editions de Physique, Les Ulis, France (1994). 19. T.-J. Chuang and J. R. Rice, Acta metall. 21, 1625 (1973). 20. T.-J. Chuang, K. I. Kagawa, J. R. Rice and L. B. Sills, Acta metall. 27, 265 (1979).