Abstract
One of the most common processes used in manufacturing of multilayer ceramic packages, multilayer capacitors and large scale integration circuits is tape casting. In this process, the wet tape thickness is one of the single most determining parameters affecting the final properties of the product, and it is therefore of great interest to be able to control it. One way to control the tape thickness is to use a two doctor blade configuration in the tape casting machine. In this case, it becomes important to fix the height of the slurry in front of both doctor blades according to the desired tape thickness and casting speed (belt velocity). In the present work, the flow in both doctor blade regions of a slurry is described with a steady state momentum equation in combination with a Bingham plastic constitutive equation, and this is integrated to a closed form analytical solution for both reservoirs based on the desired wet tape thickness and casting speed. The developed model is used to investigate the impact of different material parameters and machine designs on the required slurry height. The solution is compared with experimental findings from the literature, and good agreement is found.
Introduction
Ceramics are growing in production and usage for numerous devices, like e.g. capacitors, piezoelectric actuators, gas sensors, etc., where high quality and low geometry tolerances are required. The parallel (doctor) blade process was first used in preparing ceramic tapes in the 1940s, and it has a key role in producing thin and flat ceramic tapes.
1
Tape casting is a forming method that has mainly been used in the electronics industry to produce multilayer capacitors and electronic substrates.
2
This technique is a well established process that is used to produce ceramic layers and multilayer ceramics. The general schematic of the process is illustrated in Fig. 1a. In the tape casting process, the ceramic slurry is mostly categorised as a non-Newtonian fluid with relatively high viscosity. A summary of work published regarding the rheological classification of non-Newtonian fluids and the existence of analytical/numerical models with focus on tape casting has been given before by the authors.
3
In the present study, the Bingham plastic constitutive model is used, where the material has a yield point (τy = 15 Pa in Fig. 1b), below which no flow takes place, whereas above it, the behaviour is linear and characterised by the plastic viscosity k
a two-dimensional illustration of tape casting process with two doctor blades and b example of Bingham plastic model
Generally, the fluid flow in the doctor blade region and the subsequent outflow can be analysed using Navier–Stokes equations in two dimensions assuming that flow is generated by both the viscous drag due to the peeling velocity of the substrate v0 and the static hydraulic pressure due to the height of the slurry (either H1 or H2) in the reservoir. 3 There are a few research papers in which the flow field and tape thickness were modelled analytically.4–8 However, all of them modelled the flow of the ceramic slurry for tape casting with one doctor blade only.
It should be emphasised that in the field of materials processing technology, there exists an inherent link between materials structure, processing conditions and final properties of the part. This is also very much the case for the present work. However, since we deal with a multistep processing route of manufacturing (tape casting followed by firing or drying and finally sintering), there exist some inherent constraints on the possibilities of varying the different materials and process parameters in the tape casting process, which will be elaborated in the following.
In general, the parameters influencing tape casting are related to either the material content (i.e. the ceramic powder, solvent, dispersant, binder, plasticiser and deflocculant) or the machine configuration (i.e. the slurry height in the reservoir, the doctor blade height, substrate velocity, the doctor blade width and so forth).2,8–10 It is moreover also well known that all these aforementioned parameters influence the tape structure and its thickness. 8 Albano and Garrido 9 investigated the influence of the slurry composition on the properties of the tapes based on the changes taking place in the rheological behaviour of the slurry. However, when it comes to the slurry composition, this is to a large extent already predetermined due to constraints from the subsequent sintering process because of the inherent relation between the material content and the final microstructures and properties of the sintered part. This means that most often, the same recipe is used for the material contents in tape casting, leaving this part of the influencing parameters relatively constrained. Thus, when trying to control important resulting parameters like tape thickness, etc. during the tape casting process itself, the main possibilities for that lie in the processing parameters rather than the slurry composition.
In general, the tape thickness is the most important parameter determining the quality of the parts produced by tape casting.8,10–12 Reaching a constant tape thickness is normally achieved industrially either using a continuous form tape caster in which the reservoir all the time is fed by slurry or by applying two doctor blades in the design of the machine (Fig. 1a), which will result in having an almost constant hydrostatic pressure during the casting process. The latter type is investigated in the present work. More specifically, the aim is to model analytically the velocity and the pressure field in both doctor blade regions combined with the Bingham plastic model for the fluid flow. Then, this model is used to predict the height of the slurry in both doctor blades based on the desired tape thickness and the belt velocity. The model is based on an approach similar to the one presented by the authors in Ref. 8, however, for two doctor blades instead of one and a Bingham fluid instead of a power law fluid. Many of the affecting parameters in the process are embedded, and they can easily be varied to evaluate their influence. The proposed models describe the flow characteristics of tape casting well. Results of the model are compared with experiments from the work by Zhang et al., 7 and good agreement is obtained.
Fluid flow analysis
In order to express the volume flow and thus the tape thickness, the velocity field equation in the doctor blade region must be developed. The pressure gradient inside the channel below the doctor blade is constant, since there is a hydrostatic pressure in front of the doctor blade, and it can be determined by the height of the slurry as shown below
From equations (2) and (3), τ is found to be
Sufficient belt velocity
When the velocity of the substrate is high enough to overcome the yield point, (τ = −A0iy+A1i>τy in 0<y<hi), the shear rates are always positive (du/dy>0), and the velocity profile below the doctor blade region can be found from equation (5)
The integration constants A1i and A2i can be found by applying the boundary conditions [u(y = 0) = 0 and u(y = hi) = v0] as below
The critical belt velocity for flow can easily be found by combining the first equation of equation (7) with equation (4) and setting v0 = vcr as well as y = hi and τ = τy, which results in
The green tape thickness can also be found from the continuity equation as
Using a similar approach upstream (i.e. for the first doctor blade region), combining equations (2) and (9) and using that ΔH2 = H2−h2, the height of the slurry behind the second doctor blade can be determined as follows
Based on the calculated value of H2, and using a similar approach upstream (i.e. for the first doctor blade region) and that ΔH1 = H1−H2, the value of the height behind the first doctor blade region can be expressed by
Insufficient belt velocity
When the velocity of the peeling belt is not sufficient to overcome the yield point, i.e. v0≤vcr, the doctor blade region will divide into two regions with a critical value of y, ycr, where below ycr the shear rates are always positive (du/dy>0), and above that are equal to zero (du/dy = 0). Consequently, the velocity profile below the doctor blade region becomes
Applying the boundary conditions in momentum and constitutive equations (equations (3) and (4)), the value of ycr and the integration constants can be determined as follows
In addition, finally, the green tape thickness can be determined by solving the continuity equation as follows
Assuming that the tape thickness and the belt velocity are known parameters, the value of the height behind the second doctor blade and consequently the first doctor blade can be determined as below
Model validation
To test the proposed model, results of modelling were compared with the experimental data from Zhang et al. 7 Based on each tape thickness, the values for the height in the second reservoir obtained from the present model in the form of A02 (A02 = ρgH2/W2), compared to that of the data from Zhang et al., 7 and summarised in Table 1. As seen, a very good agreement is found.
Comparison of present model with experimental data from Zhang et al. 7
Results and discussion
Thickness versus velocity
Figure 2 shows the effect of the substrate velocity on the green tape thickness for two different values of the second doctor blade height (h2). As seen from the figure, an increased substrate velocity results in decreasing of the tape thickness. From previous works,3–5 it was found that when the drag force is increased by increasing the substrate velocity, it becomes more dominant compared to the pressure force, which results in more stretching of the slurry over the peeling belt. Moreover, it is seen that there are two zones in both figures (separated with a dashed line), which correspond to the sufficient and insufficient belt velocity. It was found that the insufficient zone shifts to higher velocities by increasing the height of the doctor blade, which is also seen from equation (8). The other point that can be understood from the figures is that the tape thickness is always higher than the half of the doctor blade height (δ>hi/2). This can also be seen from equations (10) and (11), in which δ≤hi/2 results in zero or negative pressure gradient (Hi≤hi). Furthermore, based on equation (8), it can be found that the insufficient region will shift toward the higher velocities when increasing the slurry height, decreasing the doctor blade width and decreasing the plastic viscosity.
Variation of tape thickness by substrate velocity for a h2 = 4 mm and b h2 = 5 mm
Velocity profiles in doctor blade region
As already discussed, the flow behaviour based on the critical velocity for the belt can be categorised into two groups, i.e. sufficient and insufficient. The velocity profiles for two categories are illustrated in Fig. 3 for h2 = 4 mm and different k values. It can be seen that, when the velocity in the substrate is smaller than the critical velocity, the flow experiences the needed shear rate in some point above the belt. The aforementioned point in the velocity profile gets closer to the peeling belt by increasing the plastic viscosity k. This phenomenon can be better seen in Fig. 4, where a region with zero shear rate can be found with the belt velocities below the critical velocity. Moreover, it can be seen that for the belt velocities above the critical velocity, the shear rate values are always >0.
Velocity profiles in doctor blade region for a v0>vcr and different k and b v0≤vcr and different k; value of critical velocity in these special tests is equal to 10·5
Shear rates below doctor blade region for a v0>vcr and different k and b v0≤vcr and different k; value of critical velocity in these special tests is equal to 10·5
Slurry heights
As already discussed, the main aim of the present study is to find the slurry height behind both the doctor blades when knowing the desired tape thickness and belt velocity. When solving the equations to find the aforementioned heights, one should consider whether the belt velocity is sufficient. Moreover, the momentum and continuity equations should be solved in both doctor blade regions. Figure 5 represents the height of the slurry behind the both doctor blades for the domain presented in the box in the Fig. 1a with a dashed line. The geometrical parameters used for the data shown in Fig. 5 are summarised in Table 2. As it seen, for the constant tape thickness, if the belt velocity cannot overcome the yield point (Fig. 5b), the slurry height behind both doctor blades should be increased.
Height of slurry behind both doctor blade with a sufficient belt velocity (v0 = 12 mm s−1) and b insufficient belt velocity (v0 = 5 mm s−1)
Geometrical parameters used for modelling represented in Fig. 5
The effect of the desired tape thickness on the required slurry height behind both doctor blades is shown in Fig. 6 for both the sufficient and the insufficient condition. As seen, the slurry height behind the first doctor blade is always greater than that of the second doctor blade (H1>H2). As previously mentioned, the slurry heights for both doctor blades are higher for the insufficient condition. Moreover, it can be seen that in the case of sufficient belt velocity, the relation between the variation of the tape thickness and required slurry heights are linear.
Impact of increasing value of tape thickness on required height of slurry behind both doctor blades with a sufficient and b insufficient belt velocity
The effect of changes in the second doctor blade width W2 on the desired tape thickness is illustrated in Fig. 7a and b. Increased value of the width results in increase in the required value of the slurry height. This can be easily understood from equations (10), (11) and (15), where a higher value of the doctor blade width leads to a higher value of H2 and the resultant H1. By assuming that the value of d2+W2 is constant, then the variation in the value of W2 resembles the impact of the second reservoir size d2 on the slurry height.
Effect of second doctor blade width on both slurry heights for a sufficient and b insufficient belt velocity
Conclusions
A steady state model for the Bingham plastic constitutive behaviour of a non-Newtonian slurry was proposed and used to analyse the flow field below the doctor blade region in tape casting using two doctor blades. This proposed model was based on the continuity equation assuming incompressibility such that the decrease in the volume of the slurry in the reservoir is equal to the one that leaves the doctor blade region. The results show that based on the ability of the flow to overcome the yield stress, there are two different zones, i.e. a sufficient one and an insufficient one, in which the predicted values for the slurry height and velocity profiles are totally different. The region with the insufficient belt velocity shifts toward the higher velocities by increasing the value of the critical velocity, i.e. increasing the doctor blade hi, increasing the slurry height behind the doctor blade Hi, decreasing the doctor blade width Wi and the plastic viscosity k). Moreover, the tape thickness is always smaller than the half of the doctor blade height (δ>hi/2), no matter what belt velocity is used.
The results show that when the belt velocity is not high enough to overcome the Bingham yield point (insufficient belt velocity), there is always a region with a zero shear rate below the doctor blade, and this region decreases its width by increasing the plastic viscosity k. The required slurry height based on the desired tape thickness and the belt velocity predicted by the proposed model and the results show that in the insufficient condition, the slurry height behind the both doctor blades will increase in comparison to the sufficient condition. Moreover, the variations of the aforementioned heights are different in the sufficient and insufficient condition, showing a linear increase for the sufficient condition. On the other hand, increasing the doctor blade width Wi (or decreasing the reservoir size di) with constant velocity and tape thickness, the required slurry height behind both doctor blades will be increased. The model contains all main parameters that influence the process, and it has the flexibility to be used for different slurries, which they have different constitutive behaviour as well as different machine design.
Footnotes
Acknowledgement
The authors would like to acknowledge the support of the Scientific Research Council on Technology and Production Sciences (FTP) (contract no. 09-072888, OPTIMAC), which is part of the Danish Council for Independent Research (DFF).