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Bingham plastic fluid flow model for ceramic tape casting
Materials Science and Engineering A337 (2002) 274 /280 www.elsevier.com/locate/msea
Bingham plastic fluid flow model for ceramic tape casting Guangneng Zhang *, Yonggang Wang, Jusheng Ma Division of Electronic Materials and Packing Technology, Department of Material Science and Engineering, Tsinghua University, Beijing 100084, People’s Republic of China Received 13 September 2001; received in revised form 4 January 2002
Abstract Multilayer ceramic substrates with conductor traces are attracting a great deal of attention for their ability to increase packaging density for large-scale integration circuits. Currently one of the main methods used for the manufacture of flat ceramic packages with precise thickness control and consistency is the tape casting technique. It is crucial that the green tape thickness is controlled precisely and consistently. The fluid mechanics associated with the flow of a ceramic slurry during the tape casting process is analyzed. The flow of the slurry onto the casting surface can be modeled as a two-dimensional fluid flow through a parallel channel. The material of this study is an organic-bonded glass /alumina slurry and is modeled as a Bingham plastic fluid with a yield stress. The proposed model accurately described the fluid flow characteristics of the process, and has good agreement with experimental results. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Tape casting; Bingham plastic fluid; Fluid flow model
1. Introduction The rapid development of electronic industry has been demanding higher frequencies and speeds but lower dielectric dissipation of an electronic system. To meet this need, a novel electronic package form, multilayer low temperature cofired ceramic (LTCC) substrate, which consists of up to 61 dielectric layers [1] and buried conductor, capacitor and inductor, was consequently developed. Conventional ceramic processing technology such as dry pressing, plastic molding and extrusion are not suitable for the preparation of thin, wide, defect-free sheets with smooth surfaces, precise dimensional tolerances and adequate green strength for handling [2]. Currently, one of the common methods used for the manufacture of the intermediate products, green tapes, with precise thickness and consistency is tape casting technology. A schematic of a tape casting machine is shown in Fig. 1. The machine usually consists of three main parts: casting head, moving belt, and drive system. The ceramic powder slurry filled in the tank is spread by * Corresponding author. Tel./fax: 86-10-6277-2724 E-mail address: [email protected] (G. Zhang).
one or two doctor blades onto a moving belt driven by a stepping motor, and consequently a green tape is formed on the belt. Since a thickness of about 100 mm [3] can be achieved by tape casting technology, it is crucial that the thickness of green tapes is precisely controlled and consistently reproduced. To produce unvarying tape-cast green tapes, the fluid mechanics of the process must be considered. Most of the concerning efforts have focused on the rheological behavior of dispersions used in tape casting process. Generally, there are five basic classifications of rheological behavior [3], as shown in Fig. 2. The simplest classification is modeled by a Newtonian fluid, in which the shear stress (t ) is proportional to the shear stain rate (g): ˙ The slope of the Newtonian curve shown in Fig. 2 represents the viscosity [4]. A second classification is named Bingham plastic in which a yield stress, t0, occurs at zero strain rate, followed by a linear stress /strain relationship [4]. The third classification is represented by a pseudoplastic fluid with a curving stress /strain relationship often determined by the Ostwald de Waele power-law equation. A viscoplastic fluid with a yield stress at zero strain rate and a curving stress /strain relationship fits the fourth one. Last, the fifth type of rheological behavior is dilatant, in which the shear stress
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height between the doctor blade and the moving belt when the belt velocity is low enough [3]. As a revision to the Chou model, a fluid flow model based on pseudoplastic behavior was developed by Pitchumuni and Karbhari [6] In this model, the rheology of the slurry is described by the generalized Ostwald de Waele power-law equation given as follows: t m½g½ ˙n Fig. 1. Schematic of doctor-blade tape casting process.
where t is the shear stress, g˙ is the shear rate, m and n are constants related to the non-Newtonian characteristics. To avoid the unrealistically large apparent viscosity values occurring near the zero-strain region within the power-law equation, which may lead to erroneous results of volume flow rate, Terrones et al. [7] developed a fluid model based on the three-parameter Carreau model m(g) ˙ m0
Fig. 2. Rheological behavior of ceramic powder dispersions: (A) Newtonian, (B) Bingham plastic, (C) pseudoplastic, (D) viscoplastic, and (E) dilatant.
increases with the strain rate, with an increasing slope [4]. As shown in Fig. 3, the flow of the slurry onto the moving belt can be modeled as a two-dimensional pressure and drag flow through a parallel channel. By assuming that the fluid behaves as a Newtonian fluid and flows within the channel as a linear combination of a pressure and a drag flow, a model for predicting the thickness of the green tapes was first developed by Chou and coworkers [3,5] However, tape casting slurries, consisting of a complex multiple components, seldom act as a Newtonian fluid. This model incorrectly predicts that the tape thickness might be greater than the gap
1 [1 (lg) ˙ 2 ](1n)=2
where m is the slurry viscosity as a function of shear rate, m0, l and n are constants. Ring [4] modeled tape casting behavior by applying the Bingham plastic constitutive equation t t0 mg˙ where t0 is the Bingham yield stress, m is the viscosity. Both t0 and m are constants for a certain slurry. Ring made use of shear rate as a yield criterion to divide the velocity profile into the flow and none-flow zones [3]. However, this does not correspond with the real, that the flow velocity, u , shall be greater than zero. In this paper, an alternative Bingham plastic fluid flow model for ceramic tape casting is proposed. The analysis was divided into two cases in terms of the value of belt speed, v0. When there is not sufficient belt velocity, i.e. v0 B/vc, a critical belt velocity, a relativestagnant zone exists in which the flow velocity, u , equals to the belt velocity v0. Tape casting experiments were performed at different moving belt velocities and casting head configuration with slurries of different compositions. Rheological measurements shown that the rheologies of tape casting slurries were consistent with Bingham plastic hypothesis. The predictions of the proposed model are compared with the experimental results.
2. Analysis
Fig. 3. Gap between moving belt and static doctor blade, with nomenclature and coordinate system for analysis.
In order to determine the volume flow and thus the thickness of the tape cast green sheet, it is necessary to develop equations for appropriate velocity profile within the gap between the doctor blade and moving belt [4], i.e. the parallel channel. In a tape casting system, there
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are usually two doctor blades controlled by micrometer screw gauges and a plastic moving belt driven by a stepping motor. The front doctor blade is used to ensure a hydrostatic pressure in front of the rear one [4]. For simplification, the parallel channel is isolated, as shown in Fig. 3. The coordinate system as well as the nomenclature is also shown in Fig. 3. Since there is a hydrostatic pressure exerted in front of the rear doctor blade, the pressure difference inside the parallel channel is a constant and can be established from the height of the fluid in front of the rear doctor blade as [4,8], dp rgH C0 dx W
(1)
where C0 /rgH /W is positive constant representing the pressure difference, r is the fluid density, g is the gravitational acceleration constant, H is the height of the fluid in front of the rear doctor blade, and W is the width of the doctor blade. From the momentum equilibrium under a steady state, with the assumption of infinitely long stationary plates which is required by Couette flow, it gives [4,9] dt dy
dp
(2)
dx
where t is the shear stress. From Eqs. (1) and (2), t is given by tC0 yC1
(3)
C 1 t0 C0
(6)
h
Using the momentum equilibrium formula in Eq. (2) and Bingham plastic constitutive equation, Eq. (4), the differential equation governing the velocity, u, can be determined as follows: du dy
C0 m
y
C1 t0 m
(0ByB h)
(7)
Integrating Eq. (7), the velocity, u , as a function of y, is given by u
C0
y2
2m
C 1 t0 m
yC2
(0ByBh)
(8)
where C2 is another integral constant. The boundary conditions associated with Eq. (8) are given by u(y 0) 0
(9)
u(y h) v0
(10)
where v0 is the moving belt velocity. Applying the boundary conditions to Eq. (8), the integral constants, C1 and C2 can be determined as follows: C1 t0
v0 m h
C2 0
C0 h 2
(11) (12)
where C1 is an integral constant. For Bingham plastic fluid, the shear stress, t, is given by the constitutive equation [10]
Applying Eq. (11) to inequation (6), the criterion for sufficient belt velocity is given as follows:
du tt0 mg ˙ dy
v0 vc
(4)
where u is the velocity in the x direction. From the constitutive equation (4), the shear stress, g , must be large enough to overcome the Bingham yield stress, t0, so that the shear rate, g; ˙ is greater than zero and thus the flow indeed occurs. Different belt velocities lead to different shear rate distributions in the parallel channel. So it is useful for the analysis two conditions according to the belt velocity.
C0 h 2 2m
(13)
where vc is a critical value of the belt velocity v0. It means that if v0 /vc is true, the belt velocity is sufficient to overcome the Bingham yield stress, i.e. t /t0, and (du /dy)/0 is always true for 0 B/y B/h . The velocity profile within the parallel channel is shown in Fig. 4. Finally, the thickness of the green tape, d , can be determined by calculating the average velocity as follows:
2.1. Sufficient belt velocity When there is sufficient velocity to overcome the Bingham yield stress, the following inequations must be valid: tC0 yC1 t
(0ByB h)
(5)
where h is the height of the gap between the rear doctor blade and the moving belt. Thus it requires:
Fig. 4. Velocity profile in the parallel channel with sufficient belt velocity.
G. Zhang et al. / Materials Science and Engineering A337 (2002) 274 /280
1 d v0
h
g
udy
0
C0 h 3 h 12v0 m 2
(14)
Considering the side-flow effect during the tape casting process and weight loss of the green tape in the drying process, the thickness of the dried green tape is given by, abr C0 h 3 h C0 h 3 h dtp (15) K rtp 12v0 m 2 12v0 m 2 where dtp is the thickness of the dried green tape, a is the constant for side-flow, b is the drying weight loss factor, rtp is the density of dried green tape [3], and K /abr/rtp for simplification. It shows that the Bingham yield stress, t0, has no influence on the tape thickness in this situation. 2.2. Insufficient belt velocity When there is not sufficient belt velocity to overcome the Bingham yield stress, i.e. v0 5/vc /C0h 2/2m, a critical value of y, yc, occurs between 0 and divides the shear rate (du/dy ) distribution into two parts, a relative moving zone (du /dy /0) near the stationary doctor blade, and a relative stagnant zone (du /dy /0) near the moving belt, described by: 8
0By5 yc
277
(17)
in which C0 //dp/dx /rgH /W as discussed above, C1? an integral constant. Similarly, the velocity, u , is given by 8 <C0 2 C?1 t0 y yC?2 ; 0B y5yc u 2m (18) m : v0 ; yc By Bh where C2? is an integral constant. It means that the velocity distribution consists of two parts, one near the stationary doctor blade in which u is parabolic to y , and another one near the moving belt in which u equals to v0 constantly, as shown in Fig. 5. The boundary conditions associated with Eqs. (17) and (18) are given by t(yyc )t0
(19)
u(y0)0
(20)
u(yyc )v0
(21)
Applying the boundary conditions to Eqs. (17) and (18), the value of yc and the integral constants, C1? and C2?, can be determined as follows:
Fig. 5. Velocity profile in the parallel channel with insufficient belt velocity.
sffiffiffiffiffiffiffiffiffiffi 2v0 m yc C0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ?1 2v0 mC0 t0
(22) (23) (24)
C ?2 0
Similarly, the thickness of the green tape, d , can be determined by calculating the average velocity as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffi h yc 1 1 C0 2 2v0 C0 d y y dy udy v0 v0 2m m 0 0 sffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffi 2v0 m 1 2v0 m v0 h h (25) C0 3 C0
g
g
and the thickness of dried green tape is given by sffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffi abr 1 2v0 m 1 2v0 m dtp K h h rtp 3 C0 3 C0
(26)
in which dtp, a , b, r , rtp, and K are the same as those discussed above, respectively. To summarize, a general expression of dried green tape thickness is given with the combination of Eqs. (16) and (27) as follows: 8 sffiffiffiffiffiffiffiffiffiffi > 1 2v0 m > > > ; 0B v0 5vc h C0 h 3 > > > ; v0 vc :K 2 12v0 m where C0 /rgH /W , vc /C0h 2/2m, and K is a constant related to side-flow effect and green tape weight loss. It also shows that the Bingham yield stress, t0, has no influence on the tape thickness in this situation. Fig. 6 shows the relationship between the dried tape thickness (dtp) and belt velocity (v0). For fixed values of gap height (h), plastic viscosity of fluid (m) and pressure difference (C0), the tape thickness decreases continuously with increasing belt velocity. The tape thickness distribution, as a function of belt velocity, is divided into two parts, as described in Eq. (27). Some important points are obtained from the plot: dtp/Kh /1 when v0 / 0, dtp/Kh /2/3 when v0 /vc, and dtp/Kh approaches to
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4. Results and discussion From rheology experiments, the tape casting slurry was found to behave as a Bingham plastic fluid, in which the relationship between shear stress and shear rate was t t0 mg˙
Fig. 6. Theoretical dried tape thickness (dtp/kh ) as a function of belt velocity v0.
1/2 while v0 is much greater than vc. dtp/Kh is never greater than 1 nor smaller than 1/2 no matter what belt velocity is used.
3. Experiments Twenty four tape casting experiments were performed using the common organic formulation consisting of binder, plasticizer, dispersant, homogenizer, and solvent. Tape casting experiments were performed using a two-doctor blade tape caster. The gap between the doctor blade and the moving belt was controlled by a micrometer screw gauge which could achieve an accuracy up to 0.01 mm. The belt velocity was measured through the length scale on the belt. The rheological profiles of the slurries were measured using a narrow-gap concentric-cylinder rheometer (Model NXS-11A, Chengdu Instruments Factory, China). The two parameters within the Bingham plastic constitutive equation, t0 and m, were determined by linear fitting (linear relativity /0.999) the shear stress (t) and shear rate (g) ˙ measured by the rheometer. The shrinkage factor, K , is dependent on the formulation of the slurry. Since most of those evaporated was solvent, K is mainly decided by the percentage of solvent in the slurry formulation. In the 24 experiments, the solvent percentages were set to a constant, thus, K can be considered as a constant. For all 24 experiments, the average K was 0.7464. Thickness of the dried green tape was measured 4 days after respective tape casting experiment, when it was dried completely and removed from the belt. The maximum experimental error of this thickness measurement was about 15%, mainly due to the shrinkage occurred during removal from the belt.
where t0 varied from 12.17 to 55.29 Pa while m from 0.36 to 51.33 Pa s for the 24 slurries as a result of varying amount of different components. The characteristics of the slurries, the tape casting experiment settings, and the calculated and experimental tape thicknesses for the 24 experiments are listed in Table 1. Theoretical tape thicknesses were calculated from Eq. (27) where K was chosen to be 0.7464. It can be concluded from Table 1 that: 1) Dried tape thickness (dtp) is always smaller than the gap height (h). Only when v0 B/vc is it possible that dtp /2h/3. 2) Tape thickness decreases with increasing Bingham plastic viscosity (m ) as shown in Experiments 5, 6, 7, 10, and 11. Since it was difficult to prepare slurries with the same Bingham plastic viscosity (m ) but different yield stress (t0), whether t0 has no influence on the tape thickness was not confirmed in these experiments. Comparison between theoretical tape thickness and experimental (real) thickness is shown in Fig. 7. Most of the points occur near the straight line y /x . The correlation between theoretical and experimental thickness reaches 0.9861, which means that the theory is in good agreement with reality. Fig. 8 shows the probability distribution of the relative error between the theoretical and the experimental thicknesses for the 24 experiments. It shows that in 80% of the cases, the theoretical tape thicknesses calculated from Eq. (27) had a relative error of B/10%. The maximum relative error among all the experiments was 15.8%. The drying shrinkage factor was considered as a constant in the proposed model; however, actually, it was a variable dependent on the slurry formulation. More attention to the shrinkage effect should be given in order to verify the proposed model with more confirmation.
5. Conclusions A model for predicting the thickness of tape cast substrates using Bingham plastic slurries has been developed. Whether there is a region of zero shear rate depends on the relation between belt velocity, pressure difference, gap height, and Bingham plastic viscosity.
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Table 1 Rheological parameters, tape casting settings, and tape thickness Experiment number
t0 (Pa) m (Pa s) C0 (Pa m 1)
h (mm)
v0 (m s 1)
v0 vs. vc Theoretical dtp (mm)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
18.28 35.9 39.14 23.54 31.11 19.76 20.98 27.27 28.46 26.40 22.06 21.53 26.95 20.56 24.25 19.94 19.4 12.17 55.29 22.2 28.87 32.84 31.69 52.04
1.5 1.5 2 2.5 2.5 2.5 2.5 3 3 3 3 3 3 3 0.8 0.8 0.8 1 1.2 1.2 1.35 1.35 1.6 1.6
2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.6 4.0 3.9 2.3 1.7 8.6 1.6 1.6 4.3 5.1
v0 vc v0 vc v0 vc v0 vc v0 vc v0 B vc v0 B vc v0 B vc v0 B vc v0 vc v0 vc v0 vc v0 vc v0 vc v0 vc v0 vc v0 vc v0 B vc v0 vc v0 vc v0 vc v0 vc v0 vc v0 vc
8.16 8.93 39.42 51.33 8.81 0.36 0.67 4.08 1.24 24.76 14.18 23.75 50.24 33.53 2.362 0.59 0.62 0.73 5.88 13.03 14.76 26.43 22.35 36.19
3347.86 6649.65 6584.31 6583.41 3324.82 3457.08 3422.09 3363.15 3772.28 3955.59 3955.59 6618.99 3267.74 4582.97 2652.5 1692.61 1354.09 3375.55 1989.38 3826.9 3021.85 2926.22 1313.2 1257.99
0.603 0.638 0.788 0.995 1.116 1.608 1.646 1.691 1.953 1.254 1.578 1.354 1.174 1.234 0.321 0.322 0.316 0.501 0.469 0.451 0.524 0.514 0.601 0.599
Experimental dtp (mm)
Relative error (%)
0.653 0.654 0.744 1.033 1.327 1.557 1.502 1.4873 1.9543 1.346 1.676 1.256 1.203 1.197 0.342 0.308 0.336 0.488 0.421 0.473 0.583 0.513 0.696 0.693
7.68 2.42 5.94 3.65 15.84 3.24 9.59 13.70 0.03 6.84 5.85 7.74 2.37 3.12 6.13 4.45 5.8 2.66 11.4 4.63 10.12 0.21 13.76 13.54
Fig. 7. Comparison between theoretical tape thickness and experimental tape thickness.
Fig. 8. Probability distribution of the relative error between the theoretical and the experimental thicknesses for the 24 tape casting experiments.
When there is sufficient belt velocity, positive shear rate exists within all over the parallel channel, i.e. flow velocity increases monotonously from the doctor blade to the moving belt. On the other hand, when there is not sufficient belt velocity, shear rate equals to zero and flow velocity equals to belt velocity constantly in the zone near the moving belt. It predicts that the tape thickness decreases continuously with increasing belt velocity and ranges from one gap height to half of the gap height with the effect of side-flow and weight loss neglected. It also claims that the value of Bingham yield stress has no effect on the tape thickness. More
experiments should be performed to confirm the effects of belt velocity and Bingham yield stress on the tape thickness. Accounting for the error in the thickness measurements (maximum error is 15%), it can be concluded that the agreement between experimental and theoretical tape thickness calculated from Eq. (27) is very good. By measuring rheological parameters of the slurry and accordingly adjusting the tape casting setting, a desired tape thickness with acceptable error is possibly obtained.
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Acknowledgements The research work on LTCC forming technology in this paper was aided financially by Chinese National Science Foundation (No. 69836030).
References [1] K. Niwa, E. Horikoshi, Y. Imanaka, Recent progress in mulitlayer ceramic substrates, Multilayer Electronic Ceramic Devices, American Ceramic Society, Westerville, OH, 1999, p. 175. [2] E.P. Hyatt, Am. Ceram. Soc. Bull. 65 (4) (1986) 637 /638. [3] Alfred I.Y. Tok, Freddy Y.C. Boey, Y.C. Lam, Non-Newtonian fluid flow model for ceramic tape casting, Mater. Sci. Eng. A280 (2000) 282 /288.
[4] T.A. Ring, A model of tapecasting Bingham plastic and Newtonian fluids, Adv. Ceram. 26 (1989) 569 /576. [5] Ye T. Chou, Ya T. Ko, Man F. Yan, Fluid flow model for ceramic tape casting, J. Am. Ceram. Soc. 70 (10) (1987) C-280 /C282. [6] R. Pitchumani, V.M. Karbhari, Generalized fluid flow model for ceramic tape casting, J. Am. Ceram. Soc. 78 (9) (1995) 2497 /2503. [7] G. Terrones, P.A. Smith, T.R. Armstrong, T.J. Soltesz, Application on carreau model to tape casting fluid mechanics, J. Am. Ceram. Soc. 80 (12) (1997) 3151 /3156. [8] Li Zhaomin, Cai Guoyan, Non-Newtonian Fluid Mechanics, Petroleum University Press, Beijing, China, 1998, p. 68. [9] Li Zhaomin, Cai Guoyan, Non-Newtonian Fluid Mechanics, Petroleum University Press, Beijing, China, 1998, p. 67. [10] Li Zhaomin, Cai Guoyan, Non-Newtonian Fluid Mechanics, Petroleum University Press, Beijing, China, 1998, p. 37.
Bingham plastic fluid flow model for ceramic tape casting Guangneng Zhang *, Yonggang Wang, Jusheng Ma Division of Electronic Materials and Packing Technology, Department of Material Science and Engineering, Tsinghua University, Beijing 100084, People’s Republic of China Received 13 September 2001; received in revised form 4 January 2002
Abstract Multilayer ceramic substrates with conductor traces are attracting a great deal of attention for their ability to increase packaging density for large-scale integration circuits. Currently one of the main methods used for the manufacture of flat ceramic packages with precise thickness control and consistency is the tape casting technique. It is crucial that the green tape thickness is controlled precisely and consistently. The fluid mechanics associated with the flow of a ceramic slurry during the tape casting process is analyzed. The flow of the slurry onto the casting surface can be modeled as a two-dimensional fluid flow through a parallel channel. The material of this study is an organic-bonded glass /alumina slurry and is modeled as a Bingham plastic fluid with a yield stress. The proposed model accurately described the fluid flow characteristics of the process, and has good agreement with experimental results. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Tape casting; Bingham plastic fluid; Fluid flow model
1. Introduction The rapid development of electronic industry has been demanding higher frequencies and speeds but lower dielectric dissipation of an electronic system. To meet this need, a novel electronic package form, multilayer low temperature cofired ceramic (LTCC) substrate, which consists of up to 61 dielectric layers [1] and buried conductor, capacitor and inductor, was consequently developed. Conventional ceramic processing technology such as dry pressing, plastic molding and extrusion are not suitable for the preparation of thin, wide, defect-free sheets with smooth surfaces, precise dimensional tolerances and adequate green strength for handling [2]. Currently, one of the common methods used for the manufacture of the intermediate products, green tapes, with precise thickness and consistency is tape casting technology. A schematic of a tape casting machine is shown in Fig. 1. The machine usually consists of three main parts: casting head, moving belt, and drive system. The ceramic powder slurry filled in the tank is spread by * Corresponding author. Tel./fax: 86-10-6277-2724 E-mail address: [email protected] (G. Zhang).
one or two doctor blades onto a moving belt driven by a stepping motor, and consequently a green tape is formed on the belt. Since a thickness of about 100 mm [3] can be achieved by tape casting technology, it is crucial that the thickness of green tapes is precisely controlled and consistently reproduced. To produce unvarying tape-cast green tapes, the fluid mechanics of the process must be considered. Most of the concerning efforts have focused on the rheological behavior of dispersions used in tape casting process. Generally, there are five basic classifications of rheological behavior [3], as shown in Fig. 2. The simplest classification is modeled by a Newtonian fluid, in which the shear stress (t ) is proportional to the shear stain rate (g): ˙ The slope of the Newtonian curve shown in Fig. 2 represents the viscosity [4]. A second classification is named Bingham plastic in which a yield stress, t0, occurs at zero strain rate, followed by a linear stress /strain relationship [4]. The third classification is represented by a pseudoplastic fluid with a curving stress /strain relationship often determined by the Ostwald de Waele power-law equation. A viscoplastic fluid with a yield stress at zero strain rate and a curving stress /strain relationship fits the fourth one. Last, the fifth type of rheological behavior is dilatant, in which the shear stress
0921-5093/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 2 ) 0 0 0 4 3 - 6
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height between the doctor blade and the moving belt when the belt velocity is low enough [3]. As a revision to the Chou model, a fluid flow model based on pseudoplastic behavior was developed by Pitchumuni and Karbhari [6] In this model, the rheology of the slurry is described by the generalized Ostwald de Waele power-law equation given as follows: t m½g½ ˙n Fig. 1. Schematic of doctor-blade tape casting process.
where t is the shear stress, g˙ is the shear rate, m and n are constants related to the non-Newtonian characteristics. To avoid the unrealistically large apparent viscosity values occurring near the zero-strain region within the power-law equation, which may lead to erroneous results of volume flow rate, Terrones et al. [7] developed a fluid model based on the three-parameter Carreau model m(g) ˙ m0
Fig. 2. Rheological behavior of ceramic powder dispersions: (A) Newtonian, (B) Bingham plastic, (C) pseudoplastic, (D) viscoplastic, and (E) dilatant.
increases with the strain rate, with an increasing slope [4]. As shown in Fig. 3, the flow of the slurry onto the moving belt can be modeled as a two-dimensional pressure and drag flow through a parallel channel. By assuming that the fluid behaves as a Newtonian fluid and flows within the channel as a linear combination of a pressure and a drag flow, a model for predicting the thickness of the green tapes was first developed by Chou and coworkers [3,5] However, tape casting slurries, consisting of a complex multiple components, seldom act as a Newtonian fluid. This model incorrectly predicts that the tape thickness might be greater than the gap
1 [1 (lg) ˙ 2 ](1n)=2
where m is the slurry viscosity as a function of shear rate, m0, l and n are constants. Ring [4] modeled tape casting behavior by applying the Bingham plastic constitutive equation t t0 mg˙ where t0 is the Bingham yield stress, m is the viscosity. Both t0 and m are constants for a certain slurry. Ring made use of shear rate as a yield criterion to divide the velocity profile into the flow and none-flow zones [3]. However, this does not correspond with the real, that the flow velocity, u , shall be greater than zero. In this paper, an alternative Bingham plastic fluid flow model for ceramic tape casting is proposed. The analysis was divided into two cases in terms of the value of belt speed, v0. When there is not sufficient belt velocity, i.e. v0 B/vc, a critical belt velocity, a relativestagnant zone exists in which the flow velocity, u , equals to the belt velocity v0. Tape casting experiments were performed at different moving belt velocities and casting head configuration with slurries of different compositions. Rheological measurements shown that the rheologies of tape casting slurries were consistent with Bingham plastic hypothesis. The predictions of the proposed model are compared with the experimental results.
2. Analysis
Fig. 3. Gap between moving belt and static doctor blade, with nomenclature and coordinate system for analysis.
In order to determine the volume flow and thus the thickness of the tape cast green sheet, it is necessary to develop equations for appropriate velocity profile within the gap between the doctor blade and moving belt [4], i.e. the parallel channel. In a tape casting system, there
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are usually two doctor blades controlled by micrometer screw gauges and a plastic moving belt driven by a stepping motor. The front doctor blade is used to ensure a hydrostatic pressure in front of the rear one [4]. For simplification, the parallel channel is isolated, as shown in Fig. 3. The coordinate system as well as the nomenclature is also shown in Fig. 3. Since there is a hydrostatic pressure exerted in front of the rear doctor blade, the pressure difference inside the parallel channel is a constant and can be established from the height of the fluid in front of the rear doctor blade as [4,8], dp rgH C0 dx W
(1)
where C0 /rgH /W is positive constant representing the pressure difference, r is the fluid density, g is the gravitational acceleration constant, H is the height of the fluid in front of the rear doctor blade, and W is the width of the doctor blade. From the momentum equilibrium under a steady state, with the assumption of infinitely long stationary plates which is required by Couette flow, it gives [4,9] dt dy
dp
(2)
dx
where t is the shear stress. From Eqs. (1) and (2), t is given by tC0 yC1
(3)
C 1 t0 C0
(6)
h
Using the momentum equilibrium formula in Eq. (2) and Bingham plastic constitutive equation, Eq. (4), the differential equation governing the velocity, u, can be determined as follows: du dy
C0 m
y
C1 t0 m
(0ByB h)
(7)
Integrating Eq. (7), the velocity, u , as a function of y, is given by u
C0
y2
2m
C 1 t0 m
yC2
(0ByBh)
(8)
where C2 is another integral constant. The boundary conditions associated with Eq. (8) are given by u(y 0) 0
(9)
u(y h) v0
(10)
where v0 is the moving belt velocity. Applying the boundary conditions to Eq. (8), the integral constants, C1 and C2 can be determined as follows: C1 t0
v0 m h
C2 0
C0 h 2
(11) (12)
where C1 is an integral constant. For Bingham plastic fluid, the shear stress, t, is given by the constitutive equation [10]
Applying Eq. (11) to inequation (6), the criterion for sufficient belt velocity is given as follows:
du tt0 mg ˙ dy
v0 vc
(4)
where u is the velocity in the x direction. From the constitutive equation (4), the shear stress, g , must be large enough to overcome the Bingham yield stress, t0, so that the shear rate, g; ˙ is greater than zero and thus the flow indeed occurs. Different belt velocities lead to different shear rate distributions in the parallel channel. So it is useful for the analysis two conditions according to the belt velocity.
C0 h 2 2m
(13)
where vc is a critical value of the belt velocity v0. It means that if v0 /vc is true, the belt velocity is sufficient to overcome the Bingham yield stress, i.e. t /t0, and (du /dy)/0 is always true for 0 B/y B/h . The velocity profile within the parallel channel is shown in Fig. 4. Finally, the thickness of the green tape, d , can be determined by calculating the average velocity as follows:
2.1. Sufficient belt velocity When there is sufficient velocity to overcome the Bingham yield stress, the following inequations must be valid: tC0 yC1 t
(0ByB h)
(5)
where h is the height of the gap between the rear doctor blade and the moving belt. Thus it requires:
Fig. 4. Velocity profile in the parallel channel with sufficient belt velocity.
G. Zhang et al. / Materials Science and Engineering A337 (2002) 274 /280
1 d v0
h
g
udy
0
C0 h 3 h 12v0 m 2
(14)
Considering the side-flow effect during the tape casting process and weight loss of the green tape in the drying process, the thickness of the dried green tape is given by, abr C0 h 3 h C0 h 3 h dtp (15) K rtp 12v0 m 2 12v0 m 2 where dtp is the thickness of the dried green tape, a is the constant for side-flow, b is the drying weight loss factor, rtp is the density of dried green tape [3], and K /abr/rtp for simplification. It shows that the Bingham yield stress, t0, has no influence on the tape thickness in this situation. 2.2. Insufficient belt velocity When there is not sufficient belt velocity to overcome the Bingham yield stress, i.e. v0 5/vc /C0h 2/2m, a critical value of y, yc, occurs between 0 and divides the shear rate (du/dy ) distribution into two parts, a relative moving zone (du /dy /0) near the stationary doctor blade, and a relative stagnant zone (du /dy /0) near the moving belt, described by: 8
0By5 yc
277
(17)
in which C0 //dp/dx /rgH /W as discussed above, C1? an integral constant. Similarly, the velocity, u , is given by 8 <C0 2 C?1 t0 y yC?2 ; 0B y5yc u 2m (18) m : v0 ; yc By Bh where C2? is an integral constant. It means that the velocity distribution consists of two parts, one near the stationary doctor blade in which u is parabolic to y , and another one near the moving belt in which u equals to v0 constantly, as shown in Fig. 5. The boundary conditions associated with Eqs. (17) and (18) are given by t(yyc )t0
(19)
u(y0)0
(20)
u(yyc )v0
(21)
Applying the boundary conditions to Eqs. (17) and (18), the value of yc and the integral constants, C1? and C2?, can be determined as follows:
Fig. 5. Velocity profile in the parallel channel with insufficient belt velocity.
sffiffiffiffiffiffiffiffiffiffi 2v0 m yc C0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ?1 2v0 mC0 t0
(22) (23) (24)
C ?2 0
Similarly, the thickness of the green tape, d , can be determined by calculating the average velocity as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffi h yc 1 1 C0 2 2v0 C0 d y y dy udy v0 v0 2m m 0 0 sffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffi 2v0 m 1 2v0 m v0 h h (25) C0 3 C0
g
g
and the thickness of dried green tape is given by sffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffi abr 1 2v0 m 1 2v0 m dtp K h h rtp 3 C0 3 C0
(26)
in which dtp, a , b, r , rtp, and K are the same as those discussed above, respectively. To summarize, a general expression of dried green tape thickness is given with the combination of Eqs. (16) and (27) as follows: 8 sffiffiffiffiffiffiffiffiffiffi > 1 2v0 m > > > ; 0B v0 5vc
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G. Zhang et al. / Materials Science and Engineering A337 (2002) 274 /280
4. Results and discussion From rheology experiments, the tape casting slurry was found to behave as a Bingham plastic fluid, in which the relationship between shear stress and shear rate was t t0 mg˙
Fig. 6. Theoretical dried tape thickness (dtp/kh ) as a function of belt velocity v0.
1/2 while v0 is much greater than vc. dtp/Kh is never greater than 1 nor smaller than 1/2 no matter what belt velocity is used.
3. Experiments Twenty four tape casting experiments were performed using the common organic formulation consisting of binder, plasticizer, dispersant, homogenizer, and solvent. Tape casting experiments were performed using a two-doctor blade tape caster. The gap between the doctor blade and the moving belt was controlled by a micrometer screw gauge which could achieve an accuracy up to 0.01 mm. The belt velocity was measured through the length scale on the belt. The rheological profiles of the slurries were measured using a narrow-gap concentric-cylinder rheometer (Model NXS-11A, Chengdu Instruments Factory, China). The two parameters within the Bingham plastic constitutive equation, t0 and m, were determined by linear fitting (linear relativity /0.999) the shear stress (t) and shear rate (g) ˙ measured by the rheometer. The shrinkage factor, K , is dependent on the formulation of the slurry. Since most of those evaporated was solvent, K is mainly decided by the percentage of solvent in the slurry formulation. In the 24 experiments, the solvent percentages were set to a constant, thus, K can be considered as a constant. For all 24 experiments, the average K was 0.7464. Thickness of the dried green tape was measured 4 days after respective tape casting experiment, when it was dried completely and removed from the belt. The maximum experimental error of this thickness measurement was about 15%, mainly due to the shrinkage occurred during removal from the belt.
where t0 varied from 12.17 to 55.29 Pa while m from 0.36 to 51.33 Pa s for the 24 slurries as a result of varying amount of different components. The characteristics of the slurries, the tape casting experiment settings, and the calculated and experimental tape thicknesses for the 24 experiments are listed in Table 1. Theoretical tape thicknesses were calculated from Eq. (27) where K was chosen to be 0.7464. It can be concluded from Table 1 that: 1) Dried tape thickness (dtp) is always smaller than the gap height (h). Only when v0 B/vc is it possible that dtp /2h/3. 2) Tape thickness decreases with increasing Bingham plastic viscosity (m ) as shown in Experiments 5, 6, 7, 10, and 11. Since it was difficult to prepare slurries with the same Bingham plastic viscosity (m ) but different yield stress (t0), whether t0 has no influence on the tape thickness was not confirmed in these experiments. Comparison between theoretical tape thickness and experimental (real) thickness is shown in Fig. 7. Most of the points occur near the straight line y /x . The correlation between theoretical and experimental thickness reaches 0.9861, which means that the theory is in good agreement with reality. Fig. 8 shows the probability distribution of the relative error between the theoretical and the experimental thicknesses for the 24 experiments. It shows that in 80% of the cases, the theoretical tape thicknesses calculated from Eq. (27) had a relative error of B/10%. The maximum relative error among all the experiments was 15.8%. The drying shrinkage factor was considered as a constant in the proposed model; however, actually, it was a variable dependent on the slurry formulation. More attention to the shrinkage effect should be given in order to verify the proposed model with more confirmation.
5. Conclusions A model for predicting the thickness of tape cast substrates using Bingham plastic slurries has been developed. Whether there is a region of zero shear rate depends on the relation between belt velocity, pressure difference, gap height, and Bingham plastic viscosity.
G. Zhang et al. / Materials Science and Engineering A337 (2002) 274 /280
279
Table 1 Rheological parameters, tape casting settings, and tape thickness Experiment number
t0 (Pa) m (Pa s) C0 (Pa m 1)
h (mm)
v0 (m s 1)
v0 vs. vc Theoretical dtp (mm)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
18.28 35.9 39.14 23.54 31.11 19.76 20.98 27.27 28.46 26.40 22.06 21.53 26.95 20.56 24.25 19.94 19.4 12.17 55.29 22.2 28.87 32.84 31.69 52.04
1.5 1.5 2 2.5 2.5 2.5 2.5 3 3 3 3 3 3 3 0.8 0.8 0.8 1 1.2 1.2 1.35 1.35 1.6 1.6
2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.6 4.0 3.9 2.3 1.7 8.6 1.6 1.6 4.3 5.1
v0 vc v0 vc v0 vc v0 vc v0 vc v0 B vc v0 B vc v0 B vc v0 B vc v0 vc v0 vc v0 vc v0 vc v0 vc v0 vc v0 vc v0 vc v0 B vc v0 vc v0 vc v0 vc v0 vc v0 vc v0 vc
8.16 8.93 39.42 51.33 8.81 0.36 0.67 4.08 1.24 24.76 14.18 23.75 50.24 33.53 2.362 0.59 0.62 0.73 5.88 13.03 14.76 26.43 22.35 36.19
3347.86 6649.65 6584.31 6583.41 3324.82 3457.08 3422.09 3363.15 3772.28 3955.59 3955.59 6618.99 3267.74 4582.97 2652.5 1692.61 1354.09 3375.55 1989.38 3826.9 3021.85 2926.22 1313.2 1257.99
0.603 0.638 0.788 0.995 1.116 1.608 1.646 1.691 1.953 1.254 1.578 1.354 1.174 1.234 0.321 0.322 0.316 0.501 0.469 0.451 0.524 0.514 0.601 0.599
Experimental dtp (mm)
Relative error (%)
0.653 0.654 0.744 1.033 1.327 1.557 1.502 1.4873 1.9543 1.346 1.676 1.256 1.203 1.197 0.342 0.308 0.336 0.488 0.421 0.473 0.583 0.513 0.696 0.693
7.68 2.42 5.94 3.65 15.84 3.24 9.59 13.70 0.03 6.84 5.85 7.74 2.37 3.12 6.13 4.45 5.8 2.66 11.4 4.63 10.12 0.21 13.76 13.54
Fig. 7. Comparison between theoretical tape thickness and experimental tape thickness.
Fig. 8. Probability distribution of the relative error between the theoretical and the experimental thicknesses for the 24 tape casting experiments.
When there is sufficient belt velocity, positive shear rate exists within all over the parallel channel, i.e. flow velocity increases monotonously from the doctor blade to the moving belt. On the other hand, when there is not sufficient belt velocity, shear rate equals to zero and flow velocity equals to belt velocity constantly in the zone near the moving belt. It predicts that the tape thickness decreases continuously with increasing belt velocity and ranges from one gap height to half of the gap height with the effect of side-flow and weight loss neglected. It also claims that the value of Bingham yield stress has no effect on the tape thickness. More
experiments should be performed to confirm the effects of belt velocity and Bingham yield stress on the tape thickness. Accounting for the error in the thickness measurements (maximum error is 15%), it can be concluded that the agreement between experimental and theoretical tape thickness calculated from Eq. (27) is very good. By measuring rheological parameters of the slurry and accordingly adjusting the tape casting setting, a desired tape thickness with acceptable error is possibly obtained.
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G. Zhang et al. / Materials Science and Engineering A337 (2002) 274 /280
Acknowledgements The research work on LTCC forming technology in this paper was aided financially by Chinese National Science Foundation (No. 69836030).
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