4.3 Sintering behavior of laminates
4.3.2 Sintering of a symmetric laminate
4.3.2.1 LTCC layer between elastic substrates
4.3 Sintering behavior of laminates 85
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0 3 0 0 0 3 5 0 0
0
5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0
Camber [m-1 ]
T i m e [ s ]
e x p e r i m e n t a l r e s u l t s s i m u l a t i o n
( g r e e n a l u m i n a t a p e ) s i m u l a t i o n
( d e n s e a l u m i n a s u b s t r a t e )
02468
1 0 1 2
Camber [m-1 ]
Figure 4.31: Camber experimentally measured and simulated for a LTCC laminate con-strained by a green alumina tape with an initial thickness ratio of 3. Simulation for the camber of bi-layer LTCC / dense alumina tape is also plotted.
The simulated camber for the bi-layer, dense alumina substrate / LTCC material is also indicated in gure 4.31. The same dimensions have been used for the simulation as for the experiment with the rocking arm ( 3.4.3). Experimentally, no camber has been observed. For the simulation, the camber remains extremely low. The highest camber obtained is 2.9·10−3 m−1, which corresponds to a curvature radius of 340 m.
h1
h2
Sintering Material 1: E1, ν1
Material 2: E2p, ν2p,ε&2free
)
2(z σr
)
1(z σr
h1 Material 1: E1,ν1
sintering
z
θ r
Figure 4.32: Schematic of a symmetric laminate.
σr2 =σ∞=−ε˙f ree2 E2p
1−ν2p (4.10)
However, if the elastic modulus of the outside layers has a nite value, the stress developed in the viscous laminate will be reduced. This eect is quantied by calculating the elastic response of the substrates:
εr1 =σr11−ν1
E1 (4.11)
The force balance equilibrium gives:
−2h1σr1 =h2σr2 (4.12)
The following boundary condition is assumed:
εr1 =εr2 ⇔ ε˙r1 = ˙εr2 (4.13)
Rearranging eq. 4.11, eq. 4.12 and eq. 4.13 gives:
d dt
−h2σr2
2h1
1−ν1
E1
=σr21−ν2p
E2p + ˙εf ree2 (4.14)
4.3 Sintering behavior of laminates 87
The viscous constitutive equation in the z direction is given by:
dh2 dt =h20
˙
εf ree2 −2ν2pσr2 E2p
(4.15)
where h20 is the initial thickness of the material 2.
Combining eq. 4.14 and eq. 4.15, a dierential equation of the second order is obtained:
σr2
"
ν1−1 E1
h20ε˙f ree2
2h1 +ν2p−1 E2p
# +σr22
2νp
2
E2p
1−ν1 E1
h20 2h1
+
dσr2 dt
(1−ν1)h20 2h1E1
Z
˙
εf ree2 − 2ν2pσr2 E2p
dt
= ˙εf ree2
(4.16)
h1 is taken constant for the calculation. Eq. 4.16 is numerically solved with a constant
∆t = 2s and by taking the following boundary conditions: at t = 0, σr2 = 0, ε˙f ree2 = 0. The integration procedure is the same as in Kanters' model (Appendix B).
Densication behavior of constrained laminates
Densication behavior of the laminates is measured with dierent techniques:
for the freely sintered sample, the density is calculated by measuring radial and axial strains in the sinter-forging facility,
for the laminate constrained with the green alumina tape, the density is determined by the Archimedes method for several samples heated at 840C for dierent times.
for the laminate constrained with the dense alumina substrate, the density is cal-culated by measuring the sample thickness using the rocking arm. The sample is constrained on one side but did not show any noticeable camber.
Large dierences can be observed in the densication behavior (gure 4.33). After reaching the isothermal temperature plateau of 840C, the samples are about 71 % dense after 1, 10 and 14 min, for the freely sintered sample, the laminate constrained with the green alumina tape and the laminate constrained with the dense alumina substrate, respectively. If the sample is freely sintered, the theoretical density is reached after about 8000 s, whereas if it is constrained with a dense alumina substrate, only a relative density of about 88% is reached after about 11000 s. On the other hand, when the laminate is constrained with the green alumina tape, a higher relative density is reached (∼ 95%).
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 7 0
8 0 9 0 1 0 0
Relative density [%]
T i m e [ s ] F r e e s i n t e r i n g
C o n s t r a i n e d s i n t e r i n g G r e e n a l u m i n a t a p e
C o n s t r a i n e d s i n t e r i n g D e n s e a l u m i n a s u b s t r a t e
Figure 4.33: Densication behavior for laminates freely sintered, constrained with green alumina tapes and constrained with a dense alumina substrate at 840C.
This would imply that, in this case, the sample is not fully constrained and that a limited degree of constraint exists.
Microstructure characterization
Microstructure investigation is performed on cross sections at the relative density of 86%. As can be seen in gure 4.34, the average pore area changes if the LTCC material is constrained or not, and is aected by the substrate type. It has also been observed that when the laminate is freely sintered, the average pore area remains approximately constant with density and is about the same order as the average particle area (∼ 0.4 µm2) - gure 4.35. If the sample is constrained by a green alumina tape, the average pore area is increased by a factor of 5 at 86% and 10 at relative densities larger than 92%.
If the sample is constrained by a dense alumina substrate, this factor is higher than 25 and the average pore area is about 10 µm2. Dierences in the densication behavior and the microstructure can be directly correlated: the more densication is hindered and the lower the nal density is, the larger the pores are.
In gure 4.36, the pore orientation factor is plotted as function of density for samples freely sintered, constrained with a green alumina tape and constrained with a dense alu-mina substrate. When the laalu-minate is freely sintered or constrained, the pore orientation factor kp remains the same and is equal to about 2.5. When the laminate is constrained
4.3 Sintering behavior of laminates 89
Figure 4.34: SEM micrographs of laminates a) freely sintered, b) constrained with green alumina tapes and c) constrained with a dense alumina substrate at a relative density of about 86%.
8 5 9 0 9 5 1 0 0
048
1 2
Average pore area [µm2 ]
R e l a t i v e d e n s i t y [ % ] F r e e s i n t e r i n g C o n s t r a i n e d - g r e e n a l u m i n a t a p e
C o n s t r a i n e d -
d e n s e a l u m i n a s u b s t r a t e A l u m i n a p a r t i c l e s
Figure 4.35: Average pore area as function of relative density for laminates freely sintered, constrained with green alumina tapes and constrained with a dense alumina substrate.
Average area of alumina particles is indicated by a dash line.
8 5 9 0 9 5 1 0 0
0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5
Pore orientation factor k
R e l a t i v e d e n s i t y [ % ] F r e e s i n t e r i n g C o n s t r a i n e d - g r e e n a l u m i n a t a p e
C o n s t r a i n e d -
d e n s e a l u m i n a s u b s t r a t e
Figure 4.36: Pore orientation factor kp as function of relative density for laminates freely sintered, constrained with green alumina tapes and constrained with a dense alumina substrate.
4.3 Sintering behavior of laminates 91
8 5 9 0 9 5 1 0 0
048
1 2
Particle orientation factor k p
R e l a t i v e d e n s i t y [ % ] F r e e s i n t e r i n g C o n s t r a i n e d - g r e e n a l u m i n a t a p e
C o n s t r a i n e d -
d e n s e a l u m i n a s u b s t r a t e
Figure 4.37: Particle orientation factor kp as function of relative density for laminates freely sintered, constrained with green alumina tapes and constrained with a dense alu-mina substrate.
by a dense alumina substrate, the pore orientation factor becomes lower than 1, which means that the pores are oriented perpendicularly to the plane of the laminate.
Figure 4.37 displays the particle orientation factor as function of density for the various samples. In every case, particles have an orientation factorkp larger than 1, lying between 4 and 10. When laminates are freely sintered or constrained by a dense alumina substrate, no dierence in the orientation factor is observable. If the laminate is constrained by a green alumina tape, the orientation factor is larger (two times larger at 86% of relative density).
For both constrained laminates, no gradient of porosity, average pore area and pore orientation was observed along the thickness of the laminate. The orientation factor of the particles remained also constant.
Simulation
The built-in stress in the shrinking layer is calculated using eq. 4.16.
Figure 4.38 provides stress built into the LTCC material as function of relative den-sity. A lower Young's modulus leads to a lower stress σL. To underline this eect, the normalized stressσN is dened as follows:
σN = σr2
σ∞ (4.17)
6 5 7 0 7 5 8 0 8 5 9 0 9 5 1 0 0 0 . 0
0 . 1 0 . 2 0 . 3
Stressσ r2 [MPa]
R e l a t i v e d e n s i t y [ % ] N o r a d i a l s h r i n k a g e i s a l l o w e d C o n s t r a i n e d b y a d e n s e a l u m i n a s u b s t r a t e
C o n s t r a i n e d b y a g r e e n a l u m i n a t a p e
C o n s t r a i n e d b y a t a p e w i t h a c o n s t a n t E = 1 G P a
Figure 4.38: Stress built into the LTCC material as function of relative density for a LTCC laminate constrained by (i) a rigid substrate whose Young's modulus is innite, (ii) a dense alumina substrate, (iii) a green alumina tape and (iv) a substrate whose Young's modulus is equal to 1 GPa.
whereσ∞ corresponds to the stress calculated when the laminate is perfectly constrained (eq. 4.10) and is represented by the square dots (gure 4.38).
As can be seen in gure 4.39, when the LTCC material is constrained by the dense alumina substrate, the decrease of the built-in stress is very small and represents 3 % of σ∞. When the laminate is constrained by the green alumina tape, the decrease starts from lower relative densities and is larger (about 33%). To give an order of magnitude of how built-in stress is aected by a lower value of Young's, the calculation for a Young's modulus of 1 GPa is represented by the round dots. In any case, relaxation of the stress is not observed at low relative densities (from 65 to 70 %).