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The maximum solid loading and viscosity estimation of ultra-fine BaTiO3 aqueous suspensions
Colloids and Surfaces A: Physicochem. Eng. Aspects 247 (2004) 27–34
The maximum solid loading and viscosity estimation of ultra-fine BaTiO3 aqueous suspensions Song- Yuanling, Liu- Xiaolin∗ , Chen- Jianfeng Key Lab for Nanomaterials, Ministry of Education; Research Center of the Ministry of Education for High Gravity Engineering and Technology, Beijing University of Chemical Technology, Beijing 100029, PR China Received 8 March 2004; accepted 21 August 2004 Available online 25 September 2004
Abstract The rheological behavior of ∼100 nm barium titanate (BaTiO3 ) powder aqueous suspension has been investigated over a wide range of volumetric solids loading (φ = 0.1–0.53), in which ammonium polyacrylate (NH4 PA) was used as a dispersant. The dependence of relative viscosity (ηr )–solids loading (φ) determined experimentally was compared with various existing models. The experimental results showed that the suspension apparent viscosity (η) decreased pronouncedly as the dispersant addition was increased, and the η reached a minimum as the dispersant addition exceeded 0.7wt.% (based on the dry powder weight), the viscosity increases with the solids loading and also show that the rheological behavior of the suspension was content basically with the model proposed by Liu and Mooney, and was discrepant to others. From ηr –φ relationship the maximum solid loading (φm ) was estimated to be 0.544. The fact is that yield stress (τ y ) of the suspension presented power-law dependence with φ further substantiated the validity of φm calculated. The model proposed by Liu and Mooney appears reasonably predicting the relative viscosity over a concentration range of φ = 0.3–0.5. In fact, our experimental results showed that φ might reach 0.53. © 2004 Elsevier B.V. All rights reserved. Keywords: Ultra-fine powders; Barium titanate; Suspension; Viscosity prediction
1. Introduction Barium titanate (BaTiO3 ) is a high dielectric constant material, which is widely used material in the manufacture of multilayer ceramic capacitors (MLCCs) [1,2]. With increased miniaturization and the tendency to increase the volumetric efficiency of MLCCs, the thickness of the ceramic capacitor films decreases, which means the BaTiO3 particle size ultra-fine. The current interest in using ultra-fine (nanosized) powders puts a high demand on the suspension processing and control of the forming process. Therefore, in the manufacturing process of microelectronic components, a suspension with high solids loading and desirable rheological properties are essential needed. The former requirement may ensure considerable degree of microstructure homogeneity and higher particle packing density [3,4]. The latter provides ∗
Corresponding author. E-mail address: [email protected] (L. Xiaolin).
0927-7757/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2004.08.018
critical information on the feasibility of the suspensions for specific green-shaping applications such as slip casting, tape casting, etc. In addition, the environmental and health aspects of the manufacturing process have received special attention in recent years. Therefore, suspension formulations using water as a solvent instead of organic liquids have appeared in the literatures [5–9]. Although non-aqueous solvents are faster drying and avoid hydratation of the ceramic powder, they require special precautions concerning toxicity and inflammability. Typically, organic solvent recovery systems are needed to control emissions of compounds into the atmosphere. On the other hand, an aqueous system has advantages of incombustibility, non-toxicity and low cost, associated with the large amount of experience with the use of water in similar ceramic powder process. Since the dispersion of colloidal BaTiO3 powder suspension is a key stage for green-shape process through a colloidal suspension, in the study of rheological behavior of aqueous
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suspensions, the relation between suspension viscosity (η) and solid loading (φ) has long received much attention. In short, increase in solid loading increases suspension viscosity. This η–φ relationship has been a focus of many theoretical and experimental considerations for years. Almost all of the theoretical work begins with the assumption of spherical, non-interacting rigid particles in a dilute solution. After the first equation proposed by Einstein [10] in describing η–φ behavior, a number of equations with an extended form of Einstein’s equation of were vigorously proposed [11–13] for moderate concentration. Although, for suspensions with highly concentrated solid fraction as commonly used in ceramic processing, the prediction of η–φ relation from the viscosity models aforementioned is much less accurate, the predictive capability has much improved for concentrated suspensions by taking the particle size distribution and most critically, the maximum solid loading (φm ) into account. In view of the literature, the value, φm , is by far one of the most important parameters in describing the rheological properties of colloidal suspensions [14,15] as well as in the determination of interparticle distance [16]. In this paper, we aimed to investigate rheological properties of the aqueous suspension of ultra-fine BaTiO3 powders, to relate φ with ηr and try to estimate φm . The present work hence serves as a preliminary study in conventional tape casting processes used in the ceramic forming.
2. Experimental procedure 2.1. Raw materials The commercial BaTiO3 powder (GT-BT-01) used in the work was produced by hydrothermal method, purchased from Shandong GuoTeng Functional Ceramic Material Co. Ltd., Shandong province, China. Fig. 1 is a micrograph of the BaTiO3 powders. Tables 1 and 2 show the physical characteristics and chemical composition of the powder, respectively. Ammonium polyacrylate (NH4 PA)
Table 1 Physical characteristic of BaTiO3 powder Mean particle size (nm) Surface area (m2 /g) Crystallinity
∼100 10.3 Cubic
SEM BET XRD
Table 2 Chemical compositions of the BaTiO3 powder Ba/Ti (mol ratio) Na (wt.%) Cl (wt.%) Ca (wt.%) Fe (wt.%) K (wt.%) Mg (wt.%) Sr (wt.%)
0.998 0.0010 0.0010 0.0050 0.0020 0.0010 0.0010 0.0130
was used as a dispersant with a weight average (Mw ) ∼5000. NH4 PA typically presents a linear backbone consisting of carbon–carbon bonds, with a COOH side group attached to the carbon atom. Distilled water was used as a solvent. 2.2. Experimental procedure 2.2.1. Suspension preparation The appropriate amount of NH4 PA was first mixed with distilled water, and then the powder was added to prepare suspensions. The suspensions (100 ml) were ball-milled for a period of 4 h in polyethylene bottles (500 ml) with zirconia balls by star model ball-miller (QM-ISP, Nan Jing University, China). The ball-milling time is enough to make suspensions homogenization. During milling process, pH was adjusted to about 10. 2.2.2. Rheology measurement After all suspensions with different volumetric solid loading were ball-milled as indicated above, they were poured into a concentric cylinder rheometer (Rheostress RS150L, Haake, Germany), which is able to measure the corresponding viscosity and shear stress under controlling shear rate. In this work the shear rate was applied from 0 to 300 s-1 within 3 min, then immediately from 300 to 0 s-1 . The rheological measurements of suspensions with 10, 20, 30, 40, 45, 50 and 53 vol.% were carried out at pH of 10, respectively. Steady shear measurements were performed at constant, 25 ± 1 ◦ C. Thrice measurements were made for each suspension, and each result was identical on the whole.
3. Results and discussion 3.1. Effect of dispersant additions on suspensions viscosity
Fig. 1. Powder morphology of BaTiO3 powders used in the work.
The dispersant addition is one of key factors to prepare suspension of high solid loading. In this case, the optimum
S. Yuanling et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 247 (2004) 27–34
Fig. 2. Effect of dispersant additions on the viscosity of 30 vol.% suspension.
amount of NH4 PA (on the basis of dry powder weight) was investigated by measuring the suspension η, where the solid loading of the suspensions were held a constant, 30 vol.%. Fig. 2 shows η–γ relationship when different dispersant additions were used in the BaTiO3 aqueous suspensions. We can see that the apparent viscosity of the BaTiO3 aqueous suspension decreased pronouncedly as the NH4 PA addition increased. The viscosity then reached a minimum as the dispersant addition exceeded 0.7 wt.%. 3.2. Rheology of suspension As we know that the particle size and size distribution, and the solid volume fraction may influence rheological behavior of concentrated suspension. In addition, Ba2+ ions in the suspension will influence on the stability and rheology of the suspension because of the dissolution of BaTiO3 [17], which redeposit onto the BaTiO3 surfaces to change pH of the suspension, and zeta potential of particles in the suspension. Fig. 3 shows the dependence of apparent viscosity on solid loading against shear rate. Note that the NH4 PA addition was held at 0.7 wt.% and pH at 10 for all suspensions. Firstly, we can see that the shear-thinning behavior of all suspensions over applied shear-rate, i.e., the viscosity decreases as the shear rate increases was observed and that the flow pattern of suspensions is almost perfectly Newtonian with only a slight degree of shear thinning when the solid loading of the suspension was less than φ < 0.4, and that the viscosity increased more and the shear thinning trend of suspensions become more and more when solid loading become higher. Here shear-thinning behavior can be explained as a perturbation of the suspension structure by applied shear. At low shear rates, the suspension structure is close to equilibrium, since thermal motion dominates over the viscous forces. At higher shear rates, the viscous forces affect the
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Fig. 3. Viscosity–shear rate curves for different solids loading suspension.
suspension structure more, causing the suspension structure to become distorted, hence leading to appear shear thinning. Secondly, as is shown in Fig. 3, the viscosity increases with the solids loading which mainly attribute to the increase of the interaction between particles rather than leaching Ba2+ ions. Such results have explained by some investigators [18,19] that the amount of Ba2+ ions leached from suspensions decreases with increasing pH and becomes insignificant when the initial pH > 10.0. Meanwhile the polyelectrolyte can prevent Ba2+ ions from leaching from BaTiO3 particle surface on which the polyelectrolyte was absorbed, which should be beneficial to the quality and microstructure of the BaTiO3 tapes. The viewpoint has been proved further by our experimental results, i.e. there were no Ba2+ ions detected obviously in the filtrate of a 30 vol.% suspension by sulfate precipitate method. So we can say that the leached Ba2+ ions have little influence on the particle characterization and, hence, the rheological behavior in our system. That is to say, the viscosity of concentrated suspensions depends strongly on solid loading in this work. It may be interesting to note that viscosity determined in our study is high, but the maximum solid loading is low when compared with that achievable by the microsized BaTiO3 reported by Carlos et al. and Wenjea et al. [20,21]. The difference observed could be due to the BaTiO3 particle-size used in the study of Carlos et al. and Wenjea et al. were about 0.45 and 0.8 m in average, respectively, which are about five to eight times larger than that of the powder used in this paper. As generally realized, decreasing particle size (particularly when the particle size distribution is narrow) results in an increase in the suspension viscosity, especially at low shear rates. In addition, stronger interparticle interaction is expected as the specific surface area of the particles increases given the same volumetric fraction of the solids. Fig. 4 gives the loop lines of the shear stress of suspensions with different solid loading (0.1–0.53) versus applied
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Fig. 4. Shear stress–rate curves for different solids loading.
shear rates. It is evident that all suspensions have a pseudoplastic behavior. On the other hand, the thixotropy hysteresis was found and appeared more apparent compared with micrometer particle that reported in reference [20] when the solid loading of the suspension was over 50 vol.%. This phenomenon was associated with the shear-thinning behavior of the suspensions and indicated a flocculated state of particles within the suspension. The decrease in thixotropy, viscosity and degree of shear thinning with the decrease of the solid loading of a suspension implied that the degree of powder agglomerate decreased. These results also implied that the range and magnitude of the interparticle force will have a profound influence on the suspension structure and, hence, the rheological behavior, with particles in colloidal size range (especially one dimension <100 nm). When the range of the interparticle repulsion is decrease sufficiently, the van der Waals attraction will dominate and the particles will aggregate and form flocs. In addition, the exaggerated specific surface area and higher surface activity of the ultra-fine BaTiO3 powder are suspected mainly resulting in agglomeration. 3.3. Viscosity model fitting 3.3.1. Calculated maximum solid loading value The viscosity of a suspension is strongly dependent on the solids loading, with the viscosity approaching infinity at a maximum volume fraction, φm , where φm relates to the particle concentration at which the average separation distance between the particles tends to zero and the particles pack together, making flow impossible. Fig. 5 shows the relationship of the inverse of relative viscosity of the suspension (i.e. 1/ηr ) with the solid loading (φ) at shear rate (γ) of 100 s-1 . The relative viscosity (ηr ) is defined as the suspension viscosity (η) divided by the viscosity of the liquid medium (η0 ) at given shear rate. As φ increases, the value of 1/ηr appears to decrease exponentially with φ in
Fig. 5. The relationship of the inverse of the relative viscosity to the solids loading.
a form of 1/ηr = 1.956 exp (−13.44φ) and in a correlation factor R = 0.933. The exponential curve approaches zero asymptotically as φ exceeds ∼0.53, at which the suspension ceases to flow due to the resistance force arising from the particle contact in given particle-packing configuration. The maximum solid loading φm hence indicates the volumetric ratio of the solids at which ηr approaches infinity (i.e. 1/ηr → 0). Noted that ηr = 2422.2 as φ = 0.53; therefore, φm is experimentally determined as slightly exceeding 0.53 in our model system investigated. Liu [22] recently proposed a model to predict the suspension φm over a wide range of solid loading, particle size and solvent chemistry. The model is expressed in a form of: 1 − η−1/n = aφ + b r
(1)
where n is a suspension-dependent parameter, a and b are −1/n constants that can be determined from the (1 − ηr )–φ relationship. The exponent n generally has a value of n = 2 at high shear rate but needs to be determined experimentally at low shear rate situation. −1/n Fig. 6 shows typical (1 − ηr )–φ relationship when n is taken as 1.5, 2, and 2.5, respectively. The maximum solid loading is then determined by extrapolating the linear line −1/n to 1 − ηr → 1. In the case of n = 2, the corresponding maximum solid loading is determined as φm = 0.544. The calculated φm compares quite favorably with the 1/ηr –φ relationship shown in Fig. 5. However, as n = 2.5, the calculated φm is above 0.57. The calculated φm at n = 2.5 seems approaching the upper bound of “plausible” range (Fig. 5), even if the range of “plausible” φm is not a well-defined mater and is presumably subjected to the experimental scattering. When the exponent n is 1.5, the calculated φm becomes under 0.53, which is lower than that of the experimentally attained viscosity result. The exponent n = 1.5 is hence not applicable.
S. Yuanling et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 247 (2004) 27–34
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Fig. 7. The τ 1/2 –γ 1/2 dependence for determining the suspension yield stress using the Casson’s model.
In addition, the φm calculated can be correlated with the suspension yield stress τ y for the model system to further substantiate the “righteousness” of the calculated φm range, since both τ y and φm are presumably believed to closely relate to the interparticle attractions involved in the suspension system [23]. The suspension yield stress τ y can be determined from the Casson’s equation [24]: 1/2 τ 1/2 = τy1/2 + η1/2 s γ
(2)
where τ is the apparent shear stress of the suspension. The yield stress τ y is determined from an extrapolation of the τ 1/2 –γ 1/2 linear dependence to γ 1/2 = 0, and the τ y represents the minimum stress needed to overcome before particles at rest are able to slide over adjacent particles in the liquid for the flow to occur. Fig. 7 shows the linear fit of the τ 1/2 –γ 1/2 dependence of the BaTiO3 aqueous system. Fig. 8 shows the τ y –φ relationship in ultra-fine BaTiO3 powder suspension. A pronounced increase in τ y is apparently seen as φ exceeds 0.50. The τ y determined seemed to follow a power-law relationship with the solid loading φ, as shown in Fig. 7. Poslinski et al. [25] have indicated the suspension τ y can be theoretically expressed as a function of φ as followed: φ −4 4Nεε0 kψ02 Nα 1− + (3) τy = 8πd 3 φm 4πd
−1/n
Fig. 6. The (1 − ηr )–φ relationship for determination of the maximum solid loading with different values of n: (a) n = 1.5; (b) n = 2; (c) n = 2.5.
where N is the coordination number of particles, α the Hamaker constant, d the particle size, ε the dielectric constant of the carrier fluid, ε0 the permittivity of air, ψ0 the surface potential of particles, and k the reciprocal Debye thickness of the electrostatic interaction layer. The first term on the right hand side of Eq. (3) involves the van der Waals–London attraction between particles and the second term is merely a result of the electrostatic. As φ approaches towards φm , the van der Waals attraction is expected to dominate the interparticle potentials, giving rise to an increased suspension yield stress, as indicated in Eq. (3). Since the suspension yield stress appeared
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Fig. 8. The power-law dependence of the suspension yield stress to the solid loading.
to approach infinity as the solid loading exceeds above 0.53, the power-law dependence further substantiates the validity of the previously calculated φm i.e., φm = 0.544. In general, results from the two ηr –φ and τ y –φ relationship were in reasonable agreement, which illuminated further the validity of φm = 0.544 on our ultra-fine BaTiO3 suspension.
3.3.2. Model fitting The influence of solids loading on dispersion rheology has long been an interesting and important subject for many researchers. A number of theoretical as well as empirical equations have been developed to predict the rheological behavior of concentrated suspensions. For the BaTiO3 aqueous suspensions employed presently, four models were utilized in order to predict the systems rheology. A well-known Krieger–Dougherty equation was firstly invoked to compare the experimentally determined viscosity values with that of existing model predictions [11]. The Krieger–Dougherty model has a form: −[η]φm φ ηr = 1 − φm
Fig. 9. The linear fit of ln[ηr ]–ln[1 − φ/φm ] according to equation (4).
where the n value is taken as 2 at relatively high shear rate [26]. Then Eq. (5) was expressed in a linearized form: η1/2 r −1=
[η]φφm 2(φm − φ)
(6)
Similarly, the intrinsic viscosity can be determined using 1/2 a ηr − 1 versus φ/(φm − φ) plot. Eq. (6) yields a line of slope equal to 0.272 [η] with the intercept through the origin, as illustrate in Fig. 10. The intrinsic viscosity has a value of 4.78, which is close to the value computed from Fig. 9. The third model was Mooney equation [13] given as: [η]φ ηr = exp (7) 1 − Kφ
(4)
where [η] is the intrinsic viscosity. On the basis of our experimental data the [η] value can be obtained from a linear fit of ln[ηr ]–ln[1 − φ/φm ] [26] relationship as shown in Fig. 9, and is experimentally determined as [η] = 4.3. The second one used was developed by Dabak and Yucel [12], which involve two adjusted parameters [η] and a term n. The model is given as: [η]φφm n ηr = 1 + n(φm − φ)
1/2
(5)
Fig. 10. The plot of ηr equation (6).
− 1 versus φ/(φm − φ) according to linearized
S. Yuanling et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 247 (2004) 27–34
where K is an adjustable parameter. Noted that Mooney equation is only one used in the present study that does not involve the maximum solids fraction. The [η] value is taken as 4.3 since this value was predicted in Eq. (4). The K can be obtained from Eq. (7), and is determined as 1.25 according to our experimental data. Liu [22] recently proposed a model to predict the mixture rheology. The equation has a general form: ηr = [a(φm − φ)]−n
(8)
where a is a constant, and n is a flow-dependent parameter and is also suspension specific. The term (φm − φ) is clearly defined as the effective space available for the particles to move in the matrix media. When φ ≥ φm , the effective space will reduce and the viscosity of the suspension becomes thicker and finally turns to be infinite at the point φm . The n value is taken as 2 in terms of this value predicted in Eq. (1). The 1/2 value of a can be obtained from a linear fit of ηr − (φm − φ) plot, and is determined as 1.235 for our BaTiO3 suspension system. The comparison of predictive ηr and the experimentally measured ηr were shown in Fig. 11. The number in parenthesis represents the volumetric solids loading of the particles. The model proposed by Liu and Mooney appears reasonably predicting the relative viscosity over a concentration range of φ = 0.3–0.5. Others all appear only valid in limited solids concentration and the discrepancy in ηr increases as φ > 40 vol.%. The discrepancy in the rheological prediction found among all the equations involved many probably arise from the predictive nature of the equations, as well as the sensitivity of the equations to mixture characteristics such as the polydispersity and the particle-shape effect of the powders, distribution of the carrier medium in the suspension, etc. [21,27]. In addition, the models used all assume implicitly a hard and
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same diameter sphere model in given suspension. For sterically stabilized suspension in particular, the suspension may exhibit a soft, long-range characteristic by physically adsorbing polymeric ions with certain concentration and thickness on the particle surface in given liquid medium for providing the interparticle repulsion. The sterically stabilized suspension may be theoretically treated as a hard sphere system by incorporation of an effective solid loading φeff in replace of φ in form of [28]:
φeff
δ =φ 1+ R
3 (9)
where δ is the thickness of the repulsive barrier and R the radius of the spherical particle. For precise determination of the suspension ηr , the adsorbed polymer thickness δ ought to be determined and be incorporated to the ηr –φeff plot, instead of the ηr –φ relationship used in the study. However, there is no unique measure of the thickness of a surface polymer layer due to the diffuseness of the polymer chains. Moreover, the thickness depends also on many other factors connected with the structure of the macromolecule as well as the character and charge of the adsorbent. The total surface charge of the solid, as well as the charge of the polymer chains, depends on solution pH and ionic strength [29,30]. Therefore, the discrepancy was found in the rheological prediction among all equations
4. Summary According to the varying of the ultra-fine BaTiO3 powder aqueous suspensions viscosity, the optimal dosage of the dispersant was 0.7 wt.% based on the dry powder weight. The suspension viscosity decreased with applied shear rate increase, which indicated the suspension appeared shearthinning behavior. The maximum solid loading φm = 0.544 was attained for the BaTiO3 suspensions according to a linear −1/2 (1 − ηr )–φ relationship. In addition, a theoretical, maximum solid loading was predictable for the suspension in terms of τ y –φ power-law dependence and was slightly higher than experimentally measured value (φ = 0.53), which further proved the validity of previously calculated φm . Finally, ηr –φ dependence was fitted with various existed models, but two kinds of equations proposed by Liu and Mooney agreed quite well with the dependence over a wide rage of solids loading. In other words, the use of models proposed by Liu and Mooney enable prediction of ultra-fine BaTiO3 rheology, which is in favor of shape-forming processing.
Acknowledgements
Fig. 11. The viscosity comparison of various models with experimentally measured ones.
This work was supported by the National 863 Program of China (Grant No. 2003AA302760), Fok Ying Tung Foundation (Grant No. 81063), NSF of China (Grant No. 20236020
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and No. 20325621), and the talent training program of the Beijing city (Grant No. 9558103500).
References [1] K. Yao, W. Zhu, Thin Solid Films 408 (2002) 11. [2] D.G. Schlom, J.H. Haeni, J. Lettieri, Mater. Sci. Eng. A 87 (2001) 282. [3] D.M. Liu, Ceram. Int. 23 (1997) 513. [4] D.M. Liu, Ceram. Int. 22 (1996) 101. [5] G.P. van der Beek, U. Gontermann-Gehl, E. Krafczyk, J. Eur. Ceram. Soc. 15 (1995) 741. [6] C.C. Li, J.H. Jean, J. Am. Ceram. Soc. 85 (2002) 1441. [7] Y.P. Zeng, D.L. Jiang, J. Eur. Ceram. Soc. 20 (2000) 1691. [8] Y.H. Zhang, J. Binner, J. Mater. Sci. 37 (2002) 1831. [9] Q.Q. Tan, Z.T. Zhang, Z.L. Tang, Mater. Lett. 57 (2003) 2375. [10] A. Einstein, Investigations on the Theory of Brownian Movement, Dover, New York, 1956, p. 49. [11] I.M. Krieger, T.J. Dougherty, Trans. Soc. Rheol. 3 (1959) 137. [12] T. Dabak, O. Yucel, Rheol. Acta 25 (1986) 527. [13] M. Mooney, J. Colloid Sci. 6 (1951) 162. [14] J.A. Yanez, T. Shikata, F.F. Lange, D.S. Pearson, J. Am. Ceram. Soc. 79 (19962917).
[15] T. Zhang, J.R.G. Evans, J. Eur. Ceram. Soc. 5 (1989) 165. [16] J.H. Song, J.R.G. Evans, J. Mater. Res. 9 (1994) 2386. [17] M.C. Blanco-Lopez, B. Rand, F.L. Riley, J. Eur. Ceram. Soc. 17 (1997) 281. [18] X.Y. Wang, B.I. Lee, L. Mann, Colloid Surf. A 202 (2002) 71. [19] Z.G. Shen, J.F. Chen, H.K. Zou, J.J. Yun, Colloid Interface Sci. 275 (2004) 158. [20] C.G. Yanez, H.B. Ramirez, F. Martinez, Ceram. Int. 26 (2000) 609. [21] W.J. Tseng, S.Y. Li, Mater. Sci. Eng. A 333 (2002) 314. [22] D.M. Liu, J. Mater. Sci. 35 (2000) 5503. [23] L. Bergstrom, in: R.J. Pugh, L. Bergstrom (Eds.), Surface and Colloid Chemistry in Advanced Ceramics Processing, Marcel Dekker, New York, 1994, p. 210. [24] N. Casson, Rheology of Disperse Systems, Pergaman Press, New York, 1959. [25] A.J. Poslinski, M.E. Ryan, R.K. Gupta, S.G. Seshadrl, F.J. Frechette, J. Rheol. 32 (1988) 703. [26] D.M. Liu, W.J. Tseng, J. Mater. Sci. 35 (2000) 1009. [27] L. Bergstrom, K. Shinozaki, H. Tomiyama, N. Mizutani, J. Am. Ceram. Soc. 80 (1997) 291. [28] W.M. Sigmund, N.S. Bell, L. Bergstrom, J. Am. Ceram. Soc. 83 (2000) 1557. [29] M.A. Cohen Stuart, G.J. Fleer, J. Lyklema, W. Norde, J.M.H.M. Scheutjens, Adv. Colloid Interface Sci. 34 (1991) 477. [30] S. Chibowski, M. Krupa, J. Dispersion Sci. Technol. 21 (2000) 761.
The maximum solid loading and viscosity estimation of ultra-fine BaTiO3 aqueous suspensions Song- Yuanling, Liu- Xiaolin∗ , Chen- Jianfeng Key Lab for Nanomaterials, Ministry of Education; Research Center of the Ministry of Education for High Gravity Engineering and Technology, Beijing University of Chemical Technology, Beijing 100029, PR China Received 8 March 2004; accepted 21 August 2004 Available online 25 September 2004
Abstract The rheological behavior of ∼100 nm barium titanate (BaTiO3 ) powder aqueous suspension has been investigated over a wide range of volumetric solids loading (φ = 0.1–0.53), in which ammonium polyacrylate (NH4 PA) was used as a dispersant. The dependence of relative viscosity (ηr )–solids loading (φ) determined experimentally was compared with various existing models. The experimental results showed that the suspension apparent viscosity (η) decreased pronouncedly as the dispersant addition was increased, and the η reached a minimum as the dispersant addition exceeded 0.7wt.% (based on the dry powder weight), the viscosity increases with the solids loading and also show that the rheological behavior of the suspension was content basically with the model proposed by Liu and Mooney, and was discrepant to others. From ηr –φ relationship the maximum solid loading (φm ) was estimated to be 0.544. The fact is that yield stress (τ y ) of the suspension presented power-law dependence with φ further substantiated the validity of φm calculated. The model proposed by Liu and Mooney appears reasonably predicting the relative viscosity over a concentration range of φ = 0.3–0.5. In fact, our experimental results showed that φ might reach 0.53. © 2004 Elsevier B.V. All rights reserved. Keywords: Ultra-fine powders; Barium titanate; Suspension; Viscosity prediction
1. Introduction Barium titanate (BaTiO3 ) is a high dielectric constant material, which is widely used material in the manufacture of multilayer ceramic capacitors (MLCCs) [1,2]. With increased miniaturization and the tendency to increase the volumetric efficiency of MLCCs, the thickness of the ceramic capacitor films decreases, which means the BaTiO3 particle size ultra-fine. The current interest in using ultra-fine (nanosized) powders puts a high demand on the suspension processing and control of the forming process. Therefore, in the manufacturing process of microelectronic components, a suspension with high solids loading and desirable rheological properties are essential needed. The former requirement may ensure considerable degree of microstructure homogeneity and higher particle packing density [3,4]. The latter provides ∗
Corresponding author. E-mail address: [email protected] (L. Xiaolin).
0927-7757/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2004.08.018
critical information on the feasibility of the suspensions for specific green-shaping applications such as slip casting, tape casting, etc. In addition, the environmental and health aspects of the manufacturing process have received special attention in recent years. Therefore, suspension formulations using water as a solvent instead of organic liquids have appeared in the literatures [5–9]. Although non-aqueous solvents are faster drying and avoid hydratation of the ceramic powder, they require special precautions concerning toxicity and inflammability. Typically, organic solvent recovery systems are needed to control emissions of compounds into the atmosphere. On the other hand, an aqueous system has advantages of incombustibility, non-toxicity and low cost, associated with the large amount of experience with the use of water in similar ceramic powder process. Since the dispersion of colloidal BaTiO3 powder suspension is a key stage for green-shape process through a colloidal suspension, in the study of rheological behavior of aqueous
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suspensions, the relation between suspension viscosity (η) and solid loading (φ) has long received much attention. In short, increase in solid loading increases suspension viscosity. This η–φ relationship has been a focus of many theoretical and experimental considerations for years. Almost all of the theoretical work begins with the assumption of spherical, non-interacting rigid particles in a dilute solution. After the first equation proposed by Einstein [10] in describing η–φ behavior, a number of equations with an extended form of Einstein’s equation of were vigorously proposed [11–13] for moderate concentration. Although, for suspensions with highly concentrated solid fraction as commonly used in ceramic processing, the prediction of η–φ relation from the viscosity models aforementioned is much less accurate, the predictive capability has much improved for concentrated suspensions by taking the particle size distribution and most critically, the maximum solid loading (φm ) into account. In view of the literature, the value, φm , is by far one of the most important parameters in describing the rheological properties of colloidal suspensions [14,15] as well as in the determination of interparticle distance [16]. In this paper, we aimed to investigate rheological properties of the aqueous suspension of ultra-fine BaTiO3 powders, to relate φ with ηr and try to estimate φm . The present work hence serves as a preliminary study in conventional tape casting processes used in the ceramic forming.
2. Experimental procedure 2.1. Raw materials The commercial BaTiO3 powder (GT-BT-01) used in the work was produced by hydrothermal method, purchased from Shandong GuoTeng Functional Ceramic Material Co. Ltd., Shandong province, China. Fig. 1 is a micrograph of the BaTiO3 powders. Tables 1 and 2 show the physical characteristics and chemical composition of the powder, respectively. Ammonium polyacrylate (NH4 PA)
Table 1 Physical characteristic of BaTiO3 powder Mean particle size (nm) Surface area (m2 /g) Crystallinity
∼100 10.3 Cubic
SEM BET XRD
Table 2 Chemical compositions of the BaTiO3 powder Ba/Ti (mol ratio) Na (wt.%) Cl (wt.%) Ca (wt.%) Fe (wt.%) K (wt.%) Mg (wt.%) Sr (wt.%)
0.998 0.0010 0.0010 0.0050 0.0020 0.0010 0.0010 0.0130
was used as a dispersant with a weight average (Mw ) ∼5000. NH4 PA typically presents a linear backbone consisting of carbon–carbon bonds, with a COOH side group attached to the carbon atom. Distilled water was used as a solvent. 2.2. Experimental procedure 2.2.1. Suspension preparation The appropriate amount of NH4 PA was first mixed with distilled water, and then the powder was added to prepare suspensions. The suspensions (100 ml) were ball-milled for a period of 4 h in polyethylene bottles (500 ml) with zirconia balls by star model ball-miller (QM-ISP, Nan Jing University, China). The ball-milling time is enough to make suspensions homogenization. During milling process, pH was adjusted to about 10. 2.2.2. Rheology measurement After all suspensions with different volumetric solid loading were ball-milled as indicated above, they were poured into a concentric cylinder rheometer (Rheostress RS150L, Haake, Germany), which is able to measure the corresponding viscosity and shear stress under controlling shear rate. In this work the shear rate was applied from 0 to 300 s-1 within 3 min, then immediately from 300 to 0 s-1 . The rheological measurements of suspensions with 10, 20, 30, 40, 45, 50 and 53 vol.% were carried out at pH of 10, respectively. Steady shear measurements were performed at constant, 25 ± 1 ◦ C. Thrice measurements were made for each suspension, and each result was identical on the whole.
3. Results and discussion 3.1. Effect of dispersant additions on suspensions viscosity
Fig. 1. Powder morphology of BaTiO3 powders used in the work.
The dispersant addition is one of key factors to prepare suspension of high solid loading. In this case, the optimum
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Fig. 2. Effect of dispersant additions on the viscosity of 30 vol.% suspension.
amount of NH4 PA (on the basis of dry powder weight) was investigated by measuring the suspension η, where the solid loading of the suspensions were held a constant, 30 vol.%. Fig. 2 shows η–γ relationship when different dispersant additions were used in the BaTiO3 aqueous suspensions. We can see that the apparent viscosity of the BaTiO3 aqueous suspension decreased pronouncedly as the NH4 PA addition increased. The viscosity then reached a minimum as the dispersant addition exceeded 0.7 wt.%. 3.2. Rheology of suspension As we know that the particle size and size distribution, and the solid volume fraction may influence rheological behavior of concentrated suspension. In addition, Ba2+ ions in the suspension will influence on the stability and rheology of the suspension because of the dissolution of BaTiO3 [17], which redeposit onto the BaTiO3 surfaces to change pH of the suspension, and zeta potential of particles in the suspension. Fig. 3 shows the dependence of apparent viscosity on solid loading against shear rate. Note that the NH4 PA addition was held at 0.7 wt.% and pH at 10 for all suspensions. Firstly, we can see that the shear-thinning behavior of all suspensions over applied shear-rate, i.e., the viscosity decreases as the shear rate increases was observed and that the flow pattern of suspensions is almost perfectly Newtonian with only a slight degree of shear thinning when the solid loading of the suspension was less than φ < 0.4, and that the viscosity increased more and the shear thinning trend of suspensions become more and more when solid loading become higher. Here shear-thinning behavior can be explained as a perturbation of the suspension structure by applied shear. At low shear rates, the suspension structure is close to equilibrium, since thermal motion dominates over the viscous forces. At higher shear rates, the viscous forces affect the
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Fig. 3. Viscosity–shear rate curves for different solids loading suspension.
suspension structure more, causing the suspension structure to become distorted, hence leading to appear shear thinning. Secondly, as is shown in Fig. 3, the viscosity increases with the solids loading which mainly attribute to the increase of the interaction between particles rather than leaching Ba2+ ions. Such results have explained by some investigators [18,19] that the amount of Ba2+ ions leached from suspensions decreases with increasing pH and becomes insignificant when the initial pH > 10.0. Meanwhile the polyelectrolyte can prevent Ba2+ ions from leaching from BaTiO3 particle surface on which the polyelectrolyte was absorbed, which should be beneficial to the quality and microstructure of the BaTiO3 tapes. The viewpoint has been proved further by our experimental results, i.e. there were no Ba2+ ions detected obviously in the filtrate of a 30 vol.% suspension by sulfate precipitate method. So we can say that the leached Ba2+ ions have little influence on the particle characterization and, hence, the rheological behavior in our system. That is to say, the viscosity of concentrated suspensions depends strongly on solid loading in this work. It may be interesting to note that viscosity determined in our study is high, but the maximum solid loading is low when compared with that achievable by the microsized BaTiO3 reported by Carlos et al. and Wenjea et al. [20,21]. The difference observed could be due to the BaTiO3 particle-size used in the study of Carlos et al. and Wenjea et al. were about 0.45 and 0.8 m in average, respectively, which are about five to eight times larger than that of the powder used in this paper. As generally realized, decreasing particle size (particularly when the particle size distribution is narrow) results in an increase in the suspension viscosity, especially at low shear rates. In addition, stronger interparticle interaction is expected as the specific surface area of the particles increases given the same volumetric fraction of the solids. Fig. 4 gives the loop lines of the shear stress of suspensions with different solid loading (0.1–0.53) versus applied
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Fig. 4. Shear stress–rate curves for different solids loading.
shear rates. It is evident that all suspensions have a pseudoplastic behavior. On the other hand, the thixotropy hysteresis was found and appeared more apparent compared with micrometer particle that reported in reference [20] when the solid loading of the suspension was over 50 vol.%. This phenomenon was associated with the shear-thinning behavior of the suspensions and indicated a flocculated state of particles within the suspension. The decrease in thixotropy, viscosity and degree of shear thinning with the decrease of the solid loading of a suspension implied that the degree of powder agglomerate decreased. These results also implied that the range and magnitude of the interparticle force will have a profound influence on the suspension structure and, hence, the rheological behavior, with particles in colloidal size range (especially one dimension <100 nm). When the range of the interparticle repulsion is decrease sufficiently, the van der Waals attraction will dominate and the particles will aggregate and form flocs. In addition, the exaggerated specific surface area and higher surface activity of the ultra-fine BaTiO3 powder are suspected mainly resulting in agglomeration. 3.3. Viscosity model fitting 3.3.1. Calculated maximum solid loading value The viscosity of a suspension is strongly dependent on the solids loading, with the viscosity approaching infinity at a maximum volume fraction, φm , where φm relates to the particle concentration at which the average separation distance between the particles tends to zero and the particles pack together, making flow impossible. Fig. 5 shows the relationship of the inverse of relative viscosity of the suspension (i.e. 1/ηr ) with the solid loading (φ) at shear rate (γ) of 100 s-1 . The relative viscosity (ηr ) is defined as the suspension viscosity (η) divided by the viscosity of the liquid medium (η0 ) at given shear rate. As φ increases, the value of 1/ηr appears to decrease exponentially with φ in
Fig. 5. The relationship of the inverse of the relative viscosity to the solids loading.
a form of 1/ηr = 1.956 exp (−13.44φ) and in a correlation factor R = 0.933. The exponential curve approaches zero asymptotically as φ exceeds ∼0.53, at which the suspension ceases to flow due to the resistance force arising from the particle contact in given particle-packing configuration. The maximum solid loading φm hence indicates the volumetric ratio of the solids at which ηr approaches infinity (i.e. 1/ηr → 0). Noted that ηr = 2422.2 as φ = 0.53; therefore, φm is experimentally determined as slightly exceeding 0.53 in our model system investigated. Liu [22] recently proposed a model to predict the suspension φm over a wide range of solid loading, particle size and solvent chemistry. The model is expressed in a form of: 1 − η−1/n = aφ + b r
(1)
where n is a suspension-dependent parameter, a and b are −1/n constants that can be determined from the (1 − ηr )–φ relationship. The exponent n generally has a value of n = 2 at high shear rate but needs to be determined experimentally at low shear rate situation. −1/n Fig. 6 shows typical (1 − ηr )–φ relationship when n is taken as 1.5, 2, and 2.5, respectively. The maximum solid loading is then determined by extrapolating the linear line −1/n to 1 − ηr → 1. In the case of n = 2, the corresponding maximum solid loading is determined as φm = 0.544. The calculated φm compares quite favorably with the 1/ηr –φ relationship shown in Fig. 5. However, as n = 2.5, the calculated φm is above 0.57. The calculated φm at n = 2.5 seems approaching the upper bound of “plausible” range (Fig. 5), even if the range of “plausible” φm is not a well-defined mater and is presumably subjected to the experimental scattering. When the exponent n is 1.5, the calculated φm becomes under 0.53, which is lower than that of the experimentally attained viscosity result. The exponent n = 1.5 is hence not applicable.
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Fig. 7. The τ 1/2 –γ 1/2 dependence for determining the suspension yield stress using the Casson’s model.
In addition, the φm calculated can be correlated with the suspension yield stress τ y for the model system to further substantiate the “righteousness” of the calculated φm range, since both τ y and φm are presumably believed to closely relate to the interparticle attractions involved in the suspension system [23]. The suspension yield stress τ y can be determined from the Casson’s equation [24]: 1/2 τ 1/2 = τy1/2 + η1/2 s γ
(2)
where τ is the apparent shear stress of the suspension. The yield stress τ y is determined from an extrapolation of the τ 1/2 –γ 1/2 linear dependence to γ 1/2 = 0, and the τ y represents the minimum stress needed to overcome before particles at rest are able to slide over adjacent particles in the liquid for the flow to occur. Fig. 7 shows the linear fit of the τ 1/2 –γ 1/2 dependence of the BaTiO3 aqueous system. Fig. 8 shows the τ y –φ relationship in ultra-fine BaTiO3 powder suspension. A pronounced increase in τ y is apparently seen as φ exceeds 0.50. The τ y determined seemed to follow a power-law relationship with the solid loading φ, as shown in Fig. 7. Poslinski et al. [25] have indicated the suspension τ y can be theoretically expressed as a function of φ as followed: φ −4 4Nεε0 kψ02 Nα 1− + (3) τy = 8πd 3 φm 4πd
−1/n
Fig. 6. The (1 − ηr )–φ relationship for determination of the maximum solid loading with different values of n: (a) n = 1.5; (b) n = 2; (c) n = 2.5.
where N is the coordination number of particles, α the Hamaker constant, d the particle size, ε the dielectric constant of the carrier fluid, ε0 the permittivity of air, ψ0 the surface potential of particles, and k the reciprocal Debye thickness of the electrostatic interaction layer. The first term on the right hand side of Eq. (3) involves the van der Waals–London attraction between particles and the second term is merely a result of the electrostatic. As φ approaches towards φm , the van der Waals attraction is expected to dominate the interparticle potentials, giving rise to an increased suspension yield stress, as indicated in Eq. (3). Since the suspension yield stress appeared
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Fig. 8. The power-law dependence of the suspension yield stress to the solid loading.
to approach infinity as the solid loading exceeds above 0.53, the power-law dependence further substantiates the validity of the previously calculated φm i.e., φm = 0.544. In general, results from the two ηr –φ and τ y –φ relationship were in reasonable agreement, which illuminated further the validity of φm = 0.544 on our ultra-fine BaTiO3 suspension.
3.3.2. Model fitting The influence of solids loading on dispersion rheology has long been an interesting and important subject for many researchers. A number of theoretical as well as empirical equations have been developed to predict the rheological behavior of concentrated suspensions. For the BaTiO3 aqueous suspensions employed presently, four models were utilized in order to predict the systems rheology. A well-known Krieger–Dougherty equation was firstly invoked to compare the experimentally determined viscosity values with that of existing model predictions [11]. The Krieger–Dougherty model has a form: −[η]φm φ ηr = 1 − φm
Fig. 9. The linear fit of ln[ηr ]–ln[1 − φ/φm ] according to equation (4).
where the n value is taken as 2 at relatively high shear rate [26]. Then Eq. (5) was expressed in a linearized form: η1/2 r −1=
[η]φφm 2(φm − φ)
(6)
Similarly, the intrinsic viscosity can be determined using 1/2 a ηr − 1 versus φ/(φm − φ) plot. Eq. (6) yields a line of slope equal to 0.272 [η] with the intercept through the origin, as illustrate in Fig. 10. The intrinsic viscosity has a value of 4.78, which is close to the value computed from Fig. 9. The third model was Mooney equation [13] given as: [η]φ ηr = exp (7) 1 − Kφ
(4)
where [η] is the intrinsic viscosity. On the basis of our experimental data the [η] value can be obtained from a linear fit of ln[ηr ]–ln[1 − φ/φm ] [26] relationship as shown in Fig. 9, and is experimentally determined as [η] = 4.3. The second one used was developed by Dabak and Yucel [12], which involve two adjusted parameters [η] and a term n. The model is given as: [η]φφm n ηr = 1 + n(φm − φ)
1/2
(5)
Fig. 10. The plot of ηr equation (6).
− 1 versus φ/(φm − φ) according to linearized
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where K is an adjustable parameter. Noted that Mooney equation is only one used in the present study that does not involve the maximum solids fraction. The [η] value is taken as 4.3 since this value was predicted in Eq. (4). The K can be obtained from Eq. (7), and is determined as 1.25 according to our experimental data. Liu [22] recently proposed a model to predict the mixture rheology. The equation has a general form: ηr = [a(φm − φ)]−n
(8)
where a is a constant, and n is a flow-dependent parameter and is also suspension specific. The term (φm − φ) is clearly defined as the effective space available for the particles to move in the matrix media. When φ ≥ φm , the effective space will reduce and the viscosity of the suspension becomes thicker and finally turns to be infinite at the point φm . The n value is taken as 2 in terms of this value predicted in Eq. (1). The 1/2 value of a can be obtained from a linear fit of ηr − (φm − φ) plot, and is determined as 1.235 for our BaTiO3 suspension system. The comparison of predictive ηr and the experimentally measured ηr were shown in Fig. 11. The number in parenthesis represents the volumetric solids loading of the particles. The model proposed by Liu and Mooney appears reasonably predicting the relative viscosity over a concentration range of φ = 0.3–0.5. Others all appear only valid in limited solids concentration and the discrepancy in ηr increases as φ > 40 vol.%. The discrepancy in the rheological prediction found among all the equations involved many probably arise from the predictive nature of the equations, as well as the sensitivity of the equations to mixture characteristics such as the polydispersity and the particle-shape effect of the powders, distribution of the carrier medium in the suspension, etc. [21,27]. In addition, the models used all assume implicitly a hard and
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same diameter sphere model in given suspension. For sterically stabilized suspension in particular, the suspension may exhibit a soft, long-range characteristic by physically adsorbing polymeric ions with certain concentration and thickness on the particle surface in given liquid medium for providing the interparticle repulsion. The sterically stabilized suspension may be theoretically treated as a hard sphere system by incorporation of an effective solid loading φeff in replace of φ in form of [28]:
φeff
δ =φ 1+ R
3 (9)
where δ is the thickness of the repulsive barrier and R the radius of the spherical particle. For precise determination of the suspension ηr , the adsorbed polymer thickness δ ought to be determined and be incorporated to the ηr –φeff plot, instead of the ηr –φ relationship used in the study. However, there is no unique measure of the thickness of a surface polymer layer due to the diffuseness of the polymer chains. Moreover, the thickness depends also on many other factors connected with the structure of the macromolecule as well as the character and charge of the adsorbent. The total surface charge of the solid, as well as the charge of the polymer chains, depends on solution pH and ionic strength [29,30]. Therefore, the discrepancy was found in the rheological prediction among all equations
4. Summary According to the varying of the ultra-fine BaTiO3 powder aqueous suspensions viscosity, the optimal dosage of the dispersant was 0.7 wt.% based on the dry powder weight. The suspension viscosity decreased with applied shear rate increase, which indicated the suspension appeared shearthinning behavior. The maximum solid loading φm = 0.544 was attained for the BaTiO3 suspensions according to a linear −1/2 (1 − ηr )–φ relationship. In addition, a theoretical, maximum solid loading was predictable for the suspension in terms of τ y –φ power-law dependence and was slightly higher than experimentally measured value (φ = 0.53), which further proved the validity of previously calculated φm . Finally, ηr –φ dependence was fitted with various existed models, but two kinds of equations proposed by Liu and Mooney agreed quite well with the dependence over a wide rage of solids loading. In other words, the use of models proposed by Liu and Mooney enable prediction of ultra-fine BaTiO3 rheology, which is in favor of shape-forming processing.
Acknowledgements
Fig. 11. The viscosity comparison of various models with experimentally measured ones.
This work was supported by the National 863 Program of China (Grant No. 2003AA302760), Fok Ying Tung Foundation (Grant No. 81063), NSF of China (Grant No. 20236020
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and No. 20325621), and the talent training program of the Beijing city (Grant No. 9558103500).
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